one-locus-model.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import ipywidgets as widgets\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import sympy as sym\n",
    "\n",
    "import models\n",
    "import payoffs\n",
    "import plotting\n",
    "import selection_functions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "%matplotlib inline\n",
    "plt.style.use(\"ggplot\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# One-locus model\n",
    "\n",
    "In this model both alleles of the $\\gamma$ gene are present, however only the $A$ allele of the $\\alpha$ gene is present. Formally set $x_2=1 - x_1, x_3=x_4=0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "x1 = sym.symbols(\"x1\", real=True, nonnegative=True)\n",
    "T, R, P, S, M = sym.symbols('T, R, P, S, M', real=True, positive=True)\n",
    "epsilon = sym.symbols('epsilon', real=True, positive=True)\n",
    "UGA = selection_functions.UGA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Total offspring (fitness)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = np.array([[x1], [1 - x1], [0], [0]])\n",
    "payoff_kernel = np.array([[R, S], [T, P]])\n",
    "W = models.generalized_sexual_selection(x, UGA, lambda x: x, payoff_kernel, M, 0, epsilon)\n",
    "N, = models.total_offspring(W, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 4.0 \\left(- 0.5 M + 0.5 P + \\left(- 1.0 P + 0.5 S + 0.5 T\\right) \\operatorname{UGA}{\\left(x_{1} \\right)} + \\left(0.5 P + 0.5 R - 0.5 S - 0.5 T\\right) \\operatorname{UGA}^{2}{\\left(x_{1} \\right)}\\right)$"
      ],
      "text/plain": [
       "4.0*(-0.5*M + 0.5*P + (-1.0*P + 0.5*S + 0.5*T)*UGA(x1) + (0.5*P + 0.5*R - 0.5*S - 0.5*T)*UGA(x1)**2)"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(N, UGA(x1))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After a bit of algebra we arrive at the following expression for total offspring.\n",
    "\n",
    "\\begin{align}\n",
    "   % N(x_1) =& 2.0*P*UGA(x1)**2 - 4.0*P*UGA(x1) + 2.0*P + 2.0*R*UGA(x1)**2 - 2.0*S*UGA(x1)**2 + 2.0*S*UGA(x1) - 2.0*T*UGA(x1)**2 + 2.0*T*UGA(x1) -2.0*M\\\\\n",
    "   % N(x_1) =& 2\\bigg(PU_{GA}(x_1)^2 - 2PU_{GA}(x_1) + P + RU_{GA}(x_1)^2 - SU_{GA}(x_1)^2 + SU_{GA}(x_1) - TU_{GA}(x_1)^2 + TU_{GA}(x_1) - M\\bigg) \\\\\n",
    "   % N(x_1) =& 2\\bigg(RU_{GA}(x_1)^2 + PU_{GA}(x_1)^2 - TU_{GA}(x_1)^2 - SU_{GA}(x_1)^2 + SU_{GA}(x_1) + TU_{GA}(x_1) - 2PU_{GA}(x_1) + (P - M)\\bigg) \\\\\n",
    "    N(x_1) =& 2\\bigg(\\big((R + P) - (T + S)\\big)U_{GA}(x_1)^2 + \\big((T + S) - 2P\\big)U_{GA}(x_1) + (P - M)\\bigg) \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Equilibrium total offspring\n",
    "\n",
    "Consider an equilibrium $x^*$ and define the constant equilibrium selection probability as $U_{GA}^* = U_{GA}(x^*)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "UG_star = sym.symbols(\"UG_star\")\n",
    "equilibrium_total_offspring = sym.factor(N.subs({UGA(x1): UG_star}), UG_star)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 4.0 \\left(- 0.5 M + 0.5 P + UG_{star}^{2} \\left(0.5 P + 0.5 R - 0.5 S - 0.5 T\\right) + UG_{star} \\left(- 1.0 P + 0.5 S + 0.5 T\\right)\\right)$"
      ],
      "text/plain": [
       "4.0*(-0.5*M + 0.5*P + UG_star**2*(0.5*P + 0.5*R - 0.5*S - 0.5*T) + UG_star*(-1.0*P + 0.5*S + 0.5*T))"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "equilibrium_total_offspring"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Maximizing equilibrium total offspring\n",
    "\n",
    "Find the value of the equilibrium selection probability $U_{GA}^*$ that maximizes total offspring.\n",
    "\n",
    "### Find the optimal value of $U_{GA}^*$.\n",
    "\n",
    "The number of total offspring can be written as a function of the equilibrium selection probability.\n",
    "\n",
    "$$ N(U_{GA}^*) = 2\\bigg(\\big((R + P) - (T + S)\\big)U_{GA}^{*2} + \\big((T + S) - 2P\\big)U_{GA}^* + (P - M) \\bigg)$$\n",
    "\n",
    "To find the equilibrium selection probability that maximizes the number of total offspring we need to solve the following constrained optimization problem.\n",
    "\n",
    "$$ \\max_{U_{GA}^*}\\ 2\\bigg(\\big((R + P) - (T + S)\\big)U_{GA}^{*2} + \\big((T + S) - 2P\\big)U_{GA}^* + (P - M) \\bigg) $$\n",
    "\n",
    "subject to the following inequality constraints.\n",
    "\n",
    "\\begin{align}\n",
    "    -U_{GA}^* \\le& 0 \\\\\n",
    "    U_{GA}^* - 1 \\le& 0\n",
    "\\end{align}\n",
    "\n",
    "The total offspring maximizing equilibrium value for $U_{GA}^*$ must satisfy the following first-order condition.\n",
    "\n",
    "$$ 4\\big((R + P) - (T + S)\\big)U_{GA} + 2\\big((T + S) - 2P\\big) = -\\mu_0 + \\mu_1 $$\n",
    "\n",
    "Complementary slackness conditions are\n",
    "\n",
    "\\begin{align}\n",
    "    -\\mu_0U_{GA}^* =& 0 \\\\\n",
    "    \\mu_1U_{GA}^* - 1 =& 0\n",
    "\\end{align}\n",
    "\n",
    "where $\\mu_0, \\mu_1$ are Lagrange multipliers.\n",
    "\n",
    "#### Case 1: $\\mu_0=0, \\mu_1>0$; total offspring is maximized by $U_{GA}^*=1$\n",
    "\n",
    "Substituting $U_{GA}^*=1$ into the first order condition yields\n",
    "\n",
    "$$ 4\\big((R + P) - (T + S)\\big) + 2\\big((T + S) - 2P\\big) = \\mu_1 $$\n",
    "\n",
    "which reduces to the following.\n",
    "\n",
    "$$ 4R - 2(T + S) = \\mu_1 $$\n",
    "\n",
    "The value of the Lagrange multiplier is equal to the slope of the objective at the constraint. Since the constraint is binding and the multiplier is positive it must be the case that \n",
    "\n",
    "$$ 2R > T + S $$\n",
    "\n",
    "which tells us that the joint payoff to mutual cooperation must exceed the payoff to hierarchy. Note that the total number of offspring at $N(1) = 2(R - M)$.\n",
    "\n",
    "#### Case 2: $\\mu_0>0, \\mu_1=0$; total offspring is (locally) maximized by $U_{GA}^*=0$\n",
    "\n",
    "Substituting $U_{GA}^*=0$ into the first order condition yields\n",
    "\n",
    "$$ 2\\big((T + S) - 2P\\big) = -\\mu_0 $$\n",
    "\n",
    "which reduces to the following.\n",
    "\n",
    "$$ -2\\big((T + S) - 2P\\big) = \\mu_0 $$\n",
    "\n",
    "The value of the Lagrange multiplier is equal to the slope of the objective at the constraint. Since the constraint is binding and the multiplier is positive it must be the case that \n",
    "\n",
    "$$ T + S < 2P $$\n",
    "\n",
    "which tells us that the payoff to hierarchy must be strictly less than the payoff to mutual exploitation. Note that the number of offspring at $N(0) = 2(P - M)$. Note however that because $R > P$, $U_{GA}^*=0$ can only ever be a local maximum.\n",
    "\n",
    "\n",
    "#### Case 3: $\\mu_0=0, \\mu_1=0$; total offspring is maximized by $0 < U_{GA}^* < 1$\n",
    "\n",
    "Setting the first order condition equal to zero and solving for the optimal value of $U_{GA}^*$ yield the following result.\n",
    "\n",
    "$$ \\bar{U}_{GA}^* = \\frac{1}{2}\\left(\\frac{2P - (T + S)}{(R + P) - (T + S)}\\right) $$\n",
    "\n",
    "Second order condition for this interior maximum requires that the following payoff restriction holds. \n",
    "\n",
    "$$ R + P < T + S $$\n",
    "\n",
    "The total offspring produced at this optimal equilibrium selection probability is as follows.\n",
    "\n",
    "$$ N(\\bar{U}_{GA}^*) = 2(P-M) + \\frac{1}{2}\\left(\\frac{\\big(2P - (T + S)\\big)^2}{(T + S) - (R + P)}\\right) $$\n",
    "\n",
    "An interior maximum requires that the slope of the objective at the lower bound be strictly positive and that the slope of the objective at the upper bound be strictly negative (i.e., both constraints are slack!).  This imposes the following additional restrictions on payoffs.\n",
    "\n",
    "$$ 2P < 2R < T + S $$\n",
    "\n",
    "Again, we have that the payoff to hierarchy must strictly dominate the payoff to mutual cooperation.\n",
    "\n",
    "### Discussion\n",
    "\n",
    "Now that we have gone through the maths, it is useful to characterize how the maximum value of $U_{GA}^*$ depends on model parameters.\n",
    "\n",
    "* Suppose that $2P < R + P < T + S < 2R$. In this case the unqiue global maximum value of equilibrium total offspring is $N^{**}=2(R - M)$ and is obtained at $U_{GA}^{**}=1 $.\n",
    "* Suppose that $2P < R + P < 2R < T + S$. In this case the unique global maximum value of equilibrium total offspring os $N^{**}= ???$ and is obtained at $U_{GA}^{**} = \\frac{1}{2}\\left(\\frac{2P - (T + S)}{(R + P) - (T + S)}\\right) $\n",
    "\n",
    "\n",
    "Because $R > P$, the global maximum number of offspring will either be $N(1)=2(R - M)$ or will be an interior maximum with $N(U_{GA}^*) > 2R$. If second order condition for interior maximum fails (i.e., $R + P > T + S$), then interior optimimum is actually a global minimum! In this case while the global maximum is still $U_{GA}^*=1$, there is a local maxmium at $U_{GA}^*=0$.  It might be interesting to explore the invadability of both of these points when $R + P > T + S$.\n",
    "\n",
    "#### Conditions under which $N(U_{GA}^*)$ is monotonically increasing and convex\n",
    "$N'(U_{GA}^*) > 0$ requires that \n",
    "\n",
    "$$ U_{GA}^* > \\frac{1}{2}\\left(\\frac{2P - (T + S)}{(R + P) - (T + S)}\\right) $$\n",
    "\n",
    "while $N''(U_{GA}^*) > 0$ requires that\n",
    "\n",
    "$$ (R + P) > (T + S) > 0. $$\n",
    "\n",
    "Monotonically increasing and convex $N(U_{GA}^*)$ implies global maximum occurs at $U_{GA}^*=1$ (and no local maximum at $U_{GA}^*=0$).\n",
    "\n",
    "#### Conditions under which $N(U_{GA}^*)$ is monotonically increasing and concave\n",
    "$N'(U_{GA}^*) > 0$ requires that \n",
    "\n",
    "$$ U_{GA}^* > \\frac{1}{2}\\left(\\frac{2P - (T + S)}{(R + P) - (T + S)}\\right) $$\n",
    "\n",
    "while $N''(U_{GA}^*) < 0$ requires that\n",
    "\n",
    "$$ (R + P) < (T + S)$$\n",
    "\n",
    "Monotonically increasing and concave $N(U_{GA}^*)$ implies global maximum occurs at $U_{GA}^*=1$ (and no local maximum at $U_{GA}^*=0$).\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "# solve for the interior value of UGA* that maximizes total offspring\n",
    "foc = equilibrium_total_offspring.diff(UG_star, 1)\n",
    "optimal_U_star, = sym.solve(foc, UG_star)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(P - 0.5*S - 0.5*T)/(P + R - S - T)"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimal_U_star"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "_equilibrium_total_offspring = sym.lambdify((UG_star, T, R, P, S),\n",
    "                                            equilibrium_total_offspring,\n",
    "                                            modules=\"numpy\")\n",
    "\n",
    "def plot_total_offspring(T, R, P, S):\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10,6))\n",
    "    us = np.linspace(0, 1, 100)\n",
    "    ax.plot(us, _equilibrium_total_offspring(us, T, R, P, S))\n",
    "    ax.set_title(r\"Total number of offspring\", fontsize=25)\n",
    "    ax.set_xlabel(r\"$U^*$\", fontsize=25)\n",
    "    ax.set_ylabel(r\"N\", rotation=\"horizontal\", fontsize=25)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "88fd7be33bc345cc8c1712ec3980cec9",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(FloatSlider(value=25.0, description='$T$'), FloatSlider(value=3.0, description='$R$'), F…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "# slider used to control metabolic costs\n",
    "M_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$M$\")\n",
    "\n",
    "w = widgets.interactive(plot_total_offspring, T=T_slider, R=R_slider, P=P_slider, S=S_slider, M=M_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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i5jo4olcfaWRdtOxnRiyf/Yx/9DNGjU+LzlvrGNXIzw1G1lUrrWMSo2INlOSIiEj79u3Vaf75z3+6LJuYmKiWffnllzXNX09dunRRl9+nTx+3p+/Zs6c6fbt27dya1tS71djeiaJdu3Zo3bq1y/IRERF46KGH1OfLly9Hfn6+YfG5UqdOHcTFxQEAfv/9d/U3f0YL1DaLiIhQH+/Zs0fXeTdr1gzt27d3WSY+Ph79+/dXn3/77bfFzrdXr16Ij4/3Oj5nhgwZgpiYGJdlbrvtNvVxamoqrl27dl0Z2+uM9OjRAwkJCS7nWblyZdx9991uRqu/nTt3qo8TEhJQoUIF9XlISAjuuusu9bmju9bYXm+k8G4F/kprf2HkfqIXs/o+M+i1j9pe46dBgwaafutrFl8cY/RqV3cF6vHTW3q1t+2dPXr06IGGDRsWu+z4+Hh06NABQMHdJzZv3uxO6NfRq480uy5GLJ/9jD2z+hlbRo1PAe1jVF/1e0bW1RVPxiRmxWq2Bx98UH08e/Zsp+t18+bNSElJAVAwHh8zZoxP4rNl2797e6cad6c3NTmybds29XGvXr00TdO7d2/1cVZWFvbv3697XABw/PhxzJo1CxMmTMDQoUPRu3dv9OjRw+4vKysLACAi6kVxjObPbeZKq1at1MfDhg3Dxo0bdZt3jx49NJXr2bOn+tj2YlzO2B40jdCuXbtiy1SvXl19LCLXXTgNgHohQADo0qWLpmVrLWek4m6xVdwtfW2nb9mypc7RuUev/sLI/UQrf+37zKDXPvrTTz+pjwcMGKBPcAbxxTFGr3Z1V6AeP71lxHZ8++23a15+48aN1ce2SXFP6NVHml0XI5bPfsaeWf2MLaPGp4D2Maqv+j2j6mrEmMTI9eLPBg4cqH4ReeLECSQlJTksZ3sh1h49etjtJ77w559/2t2i2d3kRnp6unrBXsDNi7ECMO1uNXl5efjjjz/U502aNNE0XYMGDRAaGoq8vDwAwJEjR9C0aVPd4jp8+DAmTJiApKQkt+59rneH6oi/tpkWTzzxBJKTkwEUZOc7duyIOnXqoEePHmjfvj1uu+02j8/SsB0saC2Xk5OD48ePo06dOk7Lu3pPD1WqVCm2TOnSpe2e2159Hij49ujMmTPqcy3fPrlTzkjFZXW7deuG0qVL4/Lly9i3bx+OHDlidzVuM+9UU0jv/sLI/aQ4/tz3mUWPfdRqtap3mwCAm266SZ/gDOCrY4we7equQD5+ekuv9t67d6/6eM6cOU4H1kUdOXJEfWx7vPKEXn2k2XXRe/nsZ65nRj9TlFHjU0DbGNWX/Z7edTVyTGLkevFnERERGDVqFN577z0ABUmQogmzzMxMfPPNN+rzBx54wKcxAsDu3bvtnrs7xrf9fODJ9KYlR4qe9hQbG6tputDQUJQrV069BZnt7c+8tWnTJvTo0QPZ2dluT2t7uyGj+GObadW7d29Mnz4dTz/9tHpbvdTUVMyYMQMzZswAANSvXx8DBgzAAw88gFq1ammet9Z2KFquuFPvjP6ZRqlSpdyepugBomgdypUrp2k+xZ1qarRLly7h6NGj6nNHHVdkZCS6deuGpUuXAig4e2Ty5MkAgFOnTuHvv/92Ob3RjOgvjNxPXPH3vs8seuyj58+ft3vNl7fCc5evjjF6tKu7Avn46S092ttqtdq14Z49ezz6WYu3yVQ9+kiz62LE8tnPXM+MfqYoo8angLYxqi/7PT3ravSYxMj14u/Gjh2rJkdWrFiBU6dOoXLlyur7CxcuVNu9SpUqms820pNtcqR8+fKoWbOmW9PbJkdCQ0M1J8MKmfazmqIbrzudmG1ZvQbmmZmZGDhwoLpBREVFYfz48Vi+fDkOHz6MS5cuITc3F1JwEVuISLHXdtCbv7WZuyZPnow9e/ZgzJgxDj+cHzp0CK+//jrq1auH5557TvNvu7W2Q3h4uN3z4trBYjH1V2eGMrtuu3fvthuEODvlzdlPa7zNCnvLyP7CqP3EjLrQ9f1M0X7InwT6McaVklw3X7hy5YqajPCGHvPwto80uy5GLJ/9jH8yanwKaBvH+bKt9aqrL8YkRq4Xf1e/fn107NgRQMGZRXPmzLF73/YnNWPGjEFoqO/Po7A9s87dxAYAbNmyRX3cpEkTt/tD084cKXpAy8zM1DytbVmt35QXZ9asWTh16pQ6z61bt6J+/fqa4/AFf2szTzRu3BizZs3CZ599hh07dmDDhg3YuHEj1q9fr/5mMDc3F6+//jouX76sZjdd0doORctFR0e7XwE/U3Sb0PpNltnfftomNypUqOA0K3zXXXepp3Zu3boVf//9N6pWrWo3fXx8PCpWrGh4zLaM7i+M2E/Mqou/MOtCmkX7W3/+5qkkHGOcKcl184UyZcrYnWaflJSE7t27mxaPN32k2XUxYvnsZ/yT2eNTX7a1XnX1xZjE7PVitrFjx2LDhg0AgJkzZ+Kpp54CAOzbtw+//PILAEBRFNx3332mxFd4MVjA/UscWK1Wu4tVd+rUye3lm/b1cVRUFCIjI9XntqfYu5KRkWG3ser1oeiHH35QHz/++OPF7ojZ2dk+P/j4W5t5IyQkBDfffDMmT56M5cuXIyMjAwsWLLA7BfbDDz9EWlpasfM6duyYpmUWbS9/Pu1Uq8jISPUq3QBw4MABTdMdPHjQqJA00Xq9kAoVKqhXFBcRLFu2zK3pjeKr/kLP/cSZQOj7HAkLC1Mfa7lImlkxly5d2u70599//92UOLQoSceYokpy3XzF9pjpL9uxp32k2XXRe/nsZ/yT2eNTX7a1XnX1xZjE7PVitgEDBtjd4acwUfLZZ5+pZTp37mza9VVsfzZveydLLZKTk3Hu3Dn1uSc3oDD13HrbDzVbt27VNE3Rco4+GNmeaqb194PHjx9XHxd3m6vCOPQ4LdJdRrWZ2cLDwzFkyBCsXr1a/dCTn5+PtWvXFjttYZbTnXLx8fElppOz3V7XrVunaRqt5YxS3J1qbDn6aY3ZyRGz+gtv9hNnAqXvK8r2g4DtgdAZ29M0fe2WW25RH+txByJPjnFaldRjDBCYdTNyXbvLdjv2ps8xktY+0uy6GLF89jP+xx/Gp75qa73q6osxiT+sF295s3+WKlUKo0aNUp9//vnnyMnJwfz589XXzLgQa6HCs+qAghtPuMP2Z0GVKlXy6Kw8U5MjhfdrBwo+9GhpANsVl5iYaPeteaEyZcqoj69cuaIpFttvHhVFKbb87NmzNc1Xb0a1mb+oV68eEhMT1eeFp9a58v3336un0bqycOFC9XFx9zcPJHfeeaf6eNWqVfjzzz9dlj9z5oyp92vPycmxO8OluIOubXJk3bp1+PPPP5Gamqp5eiOY3V94sp84Y3ZdPGX7G+Pffvut2PKFF/Y1Q7du3dTHS5YswenTp72anyfHOK1K8jEmEOtm5Lp2l+3tL7///nu7DzH+prg+0uy6GLF89jP+xx/Gp75qa73q6osxiT+sF295u38++OCD6uP//Oc/mDVrlvpFU2xsLO6++27vg/SQbRLKnYtV79q1y+5OO2PGjLE7y1grU5Mj9957r7rhnz17FtOmTXNZ/tdff8WiRYvU5/fff7/Dcra377K97ZkrVatWVR/b3ivekc2bN2PBggWa5qs3o9rMSO5mNG1P5dNyOlVmZibefPNNl2UWLlxo9wFq9OjRbsXkz4YPH67eji4nJwcPPvigXdbVlohg3Lhxbv0+s1atWlAUBYqiePTbvaL27dtnd/ArLrlRs2ZNtGzZEkDBQfO1116z26a0JEf0roMR/YXR+4kzgdL3FWW73r/++muXZbdu3Yrly5cbHZJT9957rzqQycnJwaOPPurV/Dw5xmkViMcYrQKxbkaua3cNGzZMHbTm5uZi7NixPj2LTM8+0uy6GLF89jP+xx/Gp75qa73q6osxiT+sF295u3/Wq1cPnTt3BlBwdsaTTz6pvjdixAi3LmKq9xi7Xbt26uPNmzdrSpCcO3cOgwcPVo8TcXFxmDJlimcBiJeOHTsmANS/2bNnuzX9oEGD1GktFovMnz/fYbmDBw9KfHy8WrZatWpy4cIFh2VXrlyplitTpozs37+/2DhefPFFdZqoqCjZvXu3w3Lbtm2TuLg4uzoDkHXr1jmdd8eOHT1uH0eMaDMRkXXr1qllExISio1Da/ljx47JrbfeKsuWLZPc3FyX8/zoo4/s2jUlJcVhuYSEBLtyrtph8+bNUrZsWbVsq1atJD8/v9j5ulqnRWldx57M37aex44dc1hm2rRpduV69OghqampdmWOHz8uAwYMEADXbcOu2MbcsWNHTTG78tlnn6nzi4yMlLy8vGKnefnll9VpwsLC1MflypXTtEy962BEf2HEfmJWXTzhbv+zYsUKuzic7f+//fabVKtW7bq4nTFqH33llVfsyt13332SnZ3tdJ6XLl2SN954w+F7nhzjRLT3U0YcY4xqV3cZUTd3t113eLqujWpv2/4bgPTr10/OnTvncr7Z2dkyd+5c6dKli6Y4nNG7jzSqLlr3MyOWz37G/H7GqPFp0Xm7c+w16nODEXU1akxi1HrRur3rXc7T/dPWwoULr2s/ALJv3z635qP3GDs5Odkunrp168qBAweclv/pp5+kTp06dtN88cUXHi/f9OTImTNnpGrVqnbz6NWrl8ybN0/Wr18vy5Ytk/Hjx0tkZKTdBr1q1Sqn88zJyZFKlSqp5UNCQqRVq1bSu3dv6du3r/pn6+TJk3bLKF26tDz55JOyYsUK2bhxoyxYsECGDRsmISEhAkC6d+8uNWrU0NRJ6Z0cMaLNRIxNjhSWi42NlREjRsgHH3wgK1askJ9++knWrFkjn3zyidxxxx12derfv7/TedruiIMHD1Yf9+nTR+bPny8bNmyQxYsXy3333SehoaHq++Hh4bJz505N8w2k5EheXp706NHjug4uMTFROnfuLE2aNBFFUQSAlCpVShYvXqyWCQsLc7l8vTu9Rx55RJ3fzTffrGma3377zWEH3rlzZ03T610HI/oLI/YTs+riCXf7n7y8PGncuLFdHzd69GhZsmSJbNy4Ub755hu599571WTayJEj7drNGaP20fz8/OvWXbVq1WTKlClqzElJSfLRRx/JoEGDpHTp0k7j9OQYJ6K9nzLiGOMPH1qMqpuRyRFP17WR7f3AAw/YlS1btqw8+OCDMn/+fElOTpb169fLkiVL5M0335S7775bypQpIwCkcuXKHrZCASP6SCPq4s6YT+/ls58xv58xanzqaf1EjPvcYERdjRqTGLVezEqOeLp/Fp1HxYoV7baLdu3auZzGEb3H2CIiQ4YMsYsrNDRU+vXrJ2+99ZbMnj1bZsyYIVOmTJFWrVrZlQMgU6dO9WrZpidHREQOHz58XUbP2V9YWJgsWLCg2HkuX75cIiIiXM6rqC+//FIsFkuxMTRu3FgyMjI0d1J6J0dEjGkzXyRHtP61bdtWzp8/73SeRdt+9OjRmtph2bJlLusUqMkREZErV65c15kU/YuNjZXVq1fLgQMH1Nfi4uKczjMvL0+ioqLUsuPHj9cUsytt27ZV5zd27FjN0xXNCgOQJ598stjpjKiDiP79hRH7iVl18YQnHzB37Ngh0dHRxcY9dOhQOXr0qMv+v5DR++jAgQPdWsfOeHKMc+dYpPcxxh8+tBTSu25GJkdEPFvXRra31WqV559/Xk24a/3TMzmiVx9pRF3c2c+MWD77GXP7GaPGp47m7Q4jPjcYVVcjxiRGxWpWckTEs/2zqClTptiVnzVrVrHT2DJqjH316lW566673OrHypQpI5999pnXyzb1miOFbrzxRuzZsweTJk2yuwOBLYvFgl69emH37t0YMmRIsfPs3bs39uzZgwkTJqBly5YoV64cQkJCXE4zdOhQrFixwumti6KiojB+/Hj88ssviI2NLb5iBjKizYxSuXJlTJ8+HR06dECpUqVclk1ISMA777yDDRs2uHUP+9mzZ+O9995zeruxNm3aYNu2bejTp49bsQeSiIgILFiwAD/++CPuueceJCQkIDw8HHFxcWjVqhVee+017N+/H926dbO7UJurC23t2rVLvWhVWFgYJk2a5FWMVqvV7jecxd24YZa1AAAgAElEQVSpxpbthVkLabneiN51KKR3f+GL/cSZQOr7bLVs2RIbN25EmzZtHL5fqVIlvP/++/jyyy81XdjNaBEREVi0aBEWL16MZs2auSzbsGFDTJ8+3en7nhzj3BFIxxh3BVrdjF7X7lIUBa+88gq2b9+Ofv36FdtfNWzYEE8//bR6u0hPGdFHmlUXI5fPfsa/+Mv41BdtrVddfTEm8Zf14g099s+mTZuqj6OjozFo0CC3pjdqjB0eHo7vvvsOs2bNQsOGDV2WrVSpEp566ikcPHhQl+sRKSIm3xuuiGvXrmHTpk34/fffcfbsWURFRSE+Ph6dOnXy2aA8Pz8f27Ztw65du3Dx4kVUqFABNWvWRKdOndQLX/oTf2gzrXJycrBnzx78/vvvOHXqFC5fvozIyEhUqlQJzZs3R+PGjTV9iKlVqxb++OMPAAV3MCm8AFBubi7Wr1+P1NRUnD9/HpUrV8Ytt9xid9V6At5++21MnjwZADBkyBCnF7R677331Is03XvvvZg5c6bPYtSL0XUwor/Qaz9xV6D1fbb27duHLVu2ICMjA+XKlUOdOnXQuXNnj65U7ivHjx/H5s2bcerUKWRlZSEqKgoJCQlo0aKF3d14zBZIxxh3leS6+Up2djZ+/vlnpKWl4ezZs7BYLIiJiUHt2rXRpEkTuwss6sWoPtKMuhi9fPYzvhUI41Oj21qvuvpiTOJP68UMPXv2RFJSEgDgoYcewieffOLW9L76nHD48GH8+uuvOHXqFK5cuYKYmBjExcWhRYsWqF+/vq7L8rvkCJEWzg4+VDwRQePGjZGSkgIA+OCDDzB+/HiHZe+++24sXboUFosFBw8exI033ujLUHVREupARERE/o/jUwoUf/zxB2rXrq3eKWvHjh3q3SG1KoljbL/4WQ0ReU9rnnPq1KlqYqRUqVK45557nJbdtGkTAGDQoEEB2+GVhDoQEREREenlrbfeUhMjt9xyi9uJEaBkjrGZHCEqIQYMGIDnn3/e6f3ADxw4gKFDh+KVV15RXxs3bpzTa44cOHAAGRkZUBQFzzzzjCExG60k1IGIiIiISC8rVqzAv//9b/X5U0895fY8SuoYO9TsAIhIH6dPn8aSJUvw2muvITo6GvXq1UNMTAyuXr2KtLQ0/PXXX3blW7Vqhddff93p/Bo2bKj5bBR/VRLqQERERETkqX379uH555+H1WpFWloa9u7dq7532223ObzpQXFK6hibyRGiEsJi+d+JYJcuXcL27dsdllMUBYMHD8bnn3+OiIgIX4VHREREREQ+lpGRgWXLll33epUqVTB37lwTIvJfTI4QlRArVqzAd999h+TkZKSkpODMmTM4c+YMrl69igoVKiAhIQEdOnTAsGHD3LqFLhERERERBb7Q0FDEx8ejZ8+eeOGFFwy/C1eg4d1qiIiIiIiIiCio8YKsRERERERERBTUmBwhIiIiIiIioqDG5AgRERERERERBTUmR4iIiIiIiIgoqDE5QkRERERERERBjckRIiIiIiIiIgpqTI4QERERERERUVBjcoSIiIiIiIiIghqTI0REREREREQU1JgcISIiIiIiIqKgxuQIEREREREREQU1JkeIiIiIiIiIKKgxOUJEREREREREQY3JESIiIiIiIiIKakyOEBEREREREVFQY3KEiIiIiIiIiIIakyNEREREREREFNSYHCEiIiIiIiKioMbkCBEREREREREFNSZHiIiIiIiIiCioMTlChnn++eehKAqmT5/u8P3nnnsOiqKgW7du6ms9evSAoihITk72VZhERERkIE/GA67e41iBiIiMwOQIGeLEiRN49913UbFiRYwbN85hmV27dgEAWrRoob724osvAgAmTZoEq9Xq0bJHjx4NRVE8+mvTpo1HyzTChQsX8MEHH6Bnz56Ij49HREQEypYtizp16qBDhw547rnnkJyc7HE76e3s2bOYPXs2hg8fjsTERJQpUwbh4eGoXr06+vXrhyVLlhQ7j5o1azpcL6VLl0ZiYiImTpyIM2fO+KA27nEWd1RUFJo2bYpnn30W586dMztMpwKt3fXY1oDAX29EgcDT8YCr9/QYKwDG9yX+2IdyjGSeQD/mBNJ2DnCsQB4SIgOMHDlSAMhbb73ltEynTp2kfv36smzZMrvXu3fvLgBk7ty5Hi27U6dOUrly5ev+oqKiBIAAcPh+5cqVZcqUKR4tU2/ffPONxMbGqvECkOjoaClTpozdazExMWK1Ws0OV0REQkND7WKLiIi4Lt6ePXtKdna2w+kzMjLUcuXLl1fXSdF2iI+Pl+PHj/u4ds45i7t8+fJ2cdesWVNOnDhhdrjXCcR293ZbEwn89UYUKLwZD7h6z9uxgoixfYm/9qEcI5kj0I85gbadi3CsQJ5hcoR0d+LECQkJCZFSpUrJ2bNn3Z5+yZIlAkAaNWqka1wPPPCAAJAaNWroOt+pU6cKAJk6daou85szZ44oiiIApHPnzrJy5Uq5ePGi+v7JkydlwYIF0rdvX+nWrZsuy9QDALn55pvl448/ltTUVPX1Y8eOyX333aceQIYPH+5w+tWrV6tldu/ebffeuXPn5KGHHlLfHzFihKF1cYdt3L/99pvdexkZGTJu3Dj1/VGjRnm9PL23t0Bsd2+3NRHfrzeiYOTteMAVPcYKevclgdKHOsIxkrE4VvA9jhXIE0yOkO4KO+R+/fp5NH1OTo5UqFBBAMimTZt0i+umm24SANK3b1/d5imi7wEoPT1d/fbm/vvvL7b85cuXvV6mM/n5+ZKcnCxpaWmayicnJ7t8f+zYseoBxNG3Cq+//roAkLCwMMnJyXEYT6NGjQSAxMbGaquEm9yts8j/4o6MjJS8vLzr3rdardKsWTP12zhv6T3gCcR293ZbE/H9eiMKRt6OB1zRY6ygZ18SSH2oIxwjacexQmBs5xwrkCd4zRHSlYhg5syZAIChQ4c6LTdx4kQoioK77rrruvdKlSqFAQMGAAD+7//+T5e48vLysG/fPgBAy5YtdZmnERYtWoSsrCwAwEsvvVRs+cjISN1jSElJwTPPPINatWqhS5cuOHbsmKbpOnfu7PL9++67T328ffv2697fuXMnACAxMRGlSpW67n2LxYLWrVsDKPgdaX5+vqa4tPC0zsD/4m7WrBlCQkKue19RFDRr1gwAcP78eX0C1lEgtru32xoQ+OuNyN/pMR4weqygZ18SSH1oURwjacOxQmBt5xwrkCdCzQ6ASpZ9+/bhxIkTAID27ds7Lbd7924Aji++BgAdOnTAZ599hqSkJF3i2r9/P3Jyclwu0x8cPHhQfVyhQgWfLff06dNYsGABvvjiC+zYsUN9vXz58rrFERERoT52dNAsXG7hQcaRa9euAQCioqIcHqTcoVedC6dt1aqV0zJ//vknAKBGjRoeRmucQG13V4rb1oDAX29E/k6P8YCvxwpFudOXBHIfyjGScxwrFCgJ23lRHCuQI0yOkK42btwIoKCDqFKlitNyxQ14Cq+Ifvr0aRw8eBANGjTwKq7CzC/g39+KxMTEqI9fffVVTJ06FWFhYYYs68qVK1i2bBm++OIL/PDDD8jLywNQcLDo3bs3hg0bhjvvvNPhNwSeWL9+vfq4SZMmdu+dP39e/SbA2YFXRLBlyxYAnq9DvetsG/dNN93ksMymTZuwYcMGAMDAgQM9itsogdruxXG1rQGBv96IAoEe4wFfjxWKcqcvCeQ+lGMkexwr2Csp23lRHCuQQ2b+podKnhEjRggA6d27t9MyaWlp6m/8jh075rRc4e9KZ82a5XVc48ePFwBSqVIlr+dVlJ6/6/z111/trq5dsWJFGT16tMybN0+Xq39brVZZt26d3HvvvRIdHa0ux2KxSJcuXWTmzJl2FzbTy/nz56Vq1aoCQNq3b3/d+2vWrFFjWbt2rcN5fPLJJ2qZmTNnal62kXW2jbvohbrS09Pl9ddfV7fjG2+8UZe21XN7C9R2d6W4bU3EnPVGFGy8HQ+YMVaw5W5fEsh9KMdIHCu4UlK2c1scK5AzTI6Qrm6//XYBIGPGjHFapvAK8+XKlXM5r4SEBAEgr776qtdx3XrrrQJAunfv7vW8itL7oleLFi2SGjVqqB2y7V+TJk3k3XffdfsiYykpKfLMM89IzZo17ebXvHlzefvtt+Wvv/7SJXZH8vPzpXfv3gJAwsPDr7vKuYjI9OnT1ZgyMjLU13NycmTfvn3y5JNPisViEQDSpUsXhxfFKsoXdbaNu1KlSuot3sqWLWu3zDvuuEO3W7zpub0Fars7o2VbEzFnvREFG2/HA2aMFQp50pcEch8azGMkjhWKV1K280IcK5ArTI6Qrpo2bSoAZOLEiU7LvPDCCwIU3ILNlRYtWggAefzxx72KKT8/X83sPvPMMx7NY+HChWqnWPSv8J7pZcqUcVrm559/dmt5ubm5kpSUJOPHj5fGjRurt60r/GvUqJFcunRJc+y209aqVUueffZZ2b9/vydN4bZHH3202G8TBg8e7HCgU/Rv6NChkpmZWewyfVXn4uK2WCwybdo0t+frq+0tUNvdGS3bmohx642I/sfb8YCvxwq29OpLAqEPDeYxEscKHCtwrEBFMTlCuqpXr54AkGeffdZpmbvuuksAyJNPPulyXu3atRMAMnbsWK9iOnDggNqRLVq0yKN5zJ49W9OBwdnfunXrvKrD6dOn5bPPPpM6deqo85w0aZLbsUdGRsqUKVOuOz3QKBMnTlSX/d577zktd+ONNwoAKV26tHrQrlKlitStW1fat28vkyZNkl27dmlerq/qXBj3hAkT1NdycnIkJSVFhg4dqh4816xZ49Z8fbW9BWq7O6J1WxMxbr0R0f94Ox7w9VihkCd9SSD3ocE8RuJYYZ1b8Qfydl6IYwUqDpMjpKu2bdsKAHn44YedlqlevboAkHnz5rmcV4MGDYodWGnx1VdfqR1hamqqV/NyRO9TRl05fvy4hIWFCQBp06aNpmnS09PlpZdeUjv5wr/GjRvLG2+8ofl+8e6aPHmyuqy33nrLabmLFy+q3/pMnz5dl2X7os62cc+ZM+e6961Wq9x0000CQDp27Oj18grptb0Fars7onVbEzFvvREFG2/HA74eK4h43pcEch8azGMkjhWKV1K2cxGOFUgbJkdIV3369BEAMnjwYIfvZ2RkqB3T3r17Xc6rUqVKmjK7xZk0aZIAkJiYGLFarV7NyxFfHvhFRGrXrq0eRNy1ZcsWeeSRRyQ2NlZdD4qiyG233SaffPKJ3W9JvVHY5loOpuvXr1fLJicn67J8W0bV2TZuZ79XnTdvnlrm6NGj3lRDpdf2FqjtXpQ725qIeeuNKNh4Mx4wc6zgSV9SEvrQYB8jcazgWEnbzjlWoOIwOUK6evrppwWAtG7d2uH7P/74owCQiIgIyc3NdTqfS5cuqRd3SkpK8iqmLl26CADp1KmTV/NxxpcH/uzsbCldurQAkEGDBnk8n2vXrsmSJUukf//+Eh4ernbuYWFh0rt3b1mwYIFkZ2d7NG/bUxa1HIDeffdd9WB44cIFj5aphd51Loy7VKlScu3aNYdlLl68qH6Lpdc3Lnptb4Ha7rbc3dZEzFtvRMHGm/GAr8cK3vQlgdyHinCMVBTHCvZKwnbOsQK5g8kR0tXKlSvVzuTq1avXvV945Wdng6VCycnJAkBCQ0M1XdzJlQoVKghQ/O+WPaXHAWjPnj2arsz9yCOPqB38Dz/84PHybJ07d04+/fRT9Wr1hX9RUVEyfPhwt64YbnsAevvttzVNM2zYMAEgdevW9bQKbtOjzoVxN2/e3GW5zp07CwBp27atLrHrNeAJ1HYv5Mm2JmLeeiMKNt6MB3w5VvC2LwnUPrQQx0jOcawQ+Ns5xwrkLguIdHTrrbciNDQU165dw+7du697f9euXQCAFi1auJzPtm3bAAAtW7ZEVFSUx/GkpaXh3LlzmpZpps8//xx16tTBqFGjsHLlSly4cEF979y5c1i2bBluu+02fPzxxwCARx99FHfccYcuyy5fvjzGjh2LTZs2ITU1FS+++CLq1q2LrKwszJ8/H4cPH9Y0n6eeegrvvPMOAODdd9/FxIkTNU23c+dOAECrVq08q4AH9KhzYdzNmzd3Wa5Xr14AgK1btyI9Pd374HUSqO0OeL6tAYG/3ogChTfjAV+NFfToSwKxDy3EMZJrHCsE9nbOsQJ5gskR0lV0dLTaUXz33XfXva91wFM47dChQ72Kp7BzAwoGT/5q9+7duHr1KubNm4devXqhfPnyiI6ORlRUFGJjY9GvXz/8/PPPiIyMxBtvvIEPP/zQkDhq166NqVOn4vfff8fmzZvx8MMPaxpwHj9+HNOnTwcAWCwWTJs2DVWqVHH69/bbbwMAsrOzcejQIQDATTfdZEidiuNJnW3j1nrgFBEsW7ZMn6C9FKjtDni+rQGBv96IAok34wFfjBX06ksCrQ+1xTGSdhwrBNZ2zrECeSrU7ACo5Bk7diyWLVuGr776Cq+++ioURQEAXL58Wc32uupsjh07hi1btiAyMhIjR470KpbCAVbp0qVRv359r+ZlpJUrV2LlypVYu3Ytdu7cibS0NFy8eBEWiwVVqlRBo0aN0LVrV4wcORLVqlXzSUxt27ZF27ZtNZW1Wq12j0+dOuWyfFZWFoCCAU/htGYdeG1prbNt3MUdOBs0aIDatWvj6NGjWLp0KcaOHatLrN4I1HYHPN/WgMBfb0SBxpPxgK/GCnr1JYHWh9riGMkzHCuYg2MF8gVFRMTsIKhksVqtqFevHlJTU7FhwwZ06NDBrelffvllTJ06FWPGjMGsWbMMipKIiIiM5O14wBWOFYiISG9MjpAhFixYgKFDh6Jnz55YuXKl5umys7NRq1YtZGZm4tChQ0hISDAwSiIiIjKSp+MBVzhWICIiI/CaI2SIIUOG4Oabb8aqVavUC6Zp8dFHHyEjIwOPPfYYBztEREQBztPxgCscKxARkRFCXnzxxRfNDoJKHkVR0Lp1a1SpUgXly5dHvXr1NE23e/dutGnTBlOmTEFERITBURIREZGRPB0PuMKxAhERGYE/qyEiIiIiIiKioMaf1RARERERERFRUCuRt/I9efKkIfONi4tDRkaGIfMme2xr32Fb+xbb23fY1r5jZFvreWtOI8YH3M58i+3tO2xr32Fb+w7b2ncCZWxgi2eOEBEREREREVFQY3KEiIiIiIiIiIIakyNEREREREREFNSYHCEiIiIiIiKioMbkCBEREREREREFNSZHiIiIiIiIiCioMTlCREREREREREGNyREiIiIiIiIiCmpMjhARERERERFRUGNyhIiIiIiIiIiCGpMjRERERERERBTUmBwhIiIiIiIioqDG5AgRERERERERBTUmR4iIiIiIiIgoqDE5QkRERERERERBjckRIiIiIiIiIgpqTI4QERERERERUVBjcoSIiIiIiIiIghqTI0REREREREQU1JgcISIiIiIiIqKgxuQIEREREREREQU1JkeIiIiIiIiIKKgxOUJEREREREREQY3JESIiIiIiIiIKakyOEBEREREREVFQY3KEiIiIiIiIiIIakyNEREREREREFNSYHCEiIiIiIiKioMbkCBEREREREREFNSZHiIiIiIiIiCioMTlCREREREREREEt1OwAAODjjz/Gzp07ERMTg3feeUd9fdWqVUhKSkJISAhatmyJ4cOHmxglEREREREREZVEfpEc6dSpE3r06IEZM2aor+3btw/bt2/H22+/jbCwMFy8eNHECImIiMgRsVrNDoGIiIj8jOTnmR2C2/ziZzWJiYmIioqye+2HH35A3759ERYWBgCIiYkxIzQiIiJyQeb8C5c+fcvsMIiIiMhPyB9HcHb8MMiJY2aH4ha/OHPEkb///hsHDx7EwoULERYWhhEjRqBu3boOy65ZswZr1qwBALz55puIi4szJKbQ0FDD5k322Na+w7b2Lba377CtjZf39wmc3bYBIX2H+mVb+2J8wO3Mt9jevsO29h22te+wrY1nvXAOZz+dBigKYm+oC0tMebND0sxvkyNWqxVZWVl47bXXkJqaivfeew8fffQRFEW5rmzXrl3RtWtX9XlGRoYhMcXFxRk2b7LHtvYdtrVvsb19h21tPOuCzwFLCCLuGmxYW1erVs3jaX0xPuB25ltsb99hW/sO29p32NbGkrw8WN/7J3DxPCq88W+cy80HDGhvb8YGrvjFz2ocqVChAtq0aQNFUVC3bl1YLBZkZmaaHRYREREBkLNnIJuTobTvhpDysWaHQ0RERCaTRbOAw/uhjHwUYXXqmx2O2/w2OdK6dWvs27cPAHDy5Enk5eWhbNmyJkdFREREACCrvwUAKN37mxwJERERmc368xpI8good/SF5ZZOZofjEb/4Wc3777+PlJQUZGZm4qGHHsKgQYPQpUsXfPzxx5g4cSJCQ0Mxbtw4hz+pISIiIt+Si+chP/0ApW1nKLEVzQ6HiIiITCRHD0Hmfww0bAZlwGizw/GYXyRHJkyY4PD1xx57zMeREBERUXHkh6VAfj6UngPMDoWIiIhMJBfOwfrJm0BMBVgenAwlJMTskDzmtz+rISIiIv8jWZcgG1ZBubk9lErGXBCNiIiI/J/k5sL66ZvA5SxYxj0HJSra7JC8wuQIERERaSZrlwM5V6Hc+Q+zQyEiIiKTiAhkwb+B1IOwjHkcSo0bzA7Ja0yOEBERkSZyORuydgXQsh2UajXNDoeIiIhMIutXFVx/7M5BUFrdZnY4umByhIiIiDSRdd8DV7Jh4VkjREREQUsO7YN8/RnQtDWUvkPNDkc3TI4QERFRseTqZciPy4AmraAk1DE7HCIiIjKBnD1dcJ2RilVgue9JKJaSk1IoOTUhIiIiw8j6VUB2Jiy9B5sdChEREZlAcq7COuM1ID+v4AKspcuYHZKumBwhIiIilyQnp+D2vYktoNSub3Y4RERE5GMiApnzAXAiDZYHJkOpUt3skHTH5AgRERG5JD8lAZkXedYIERFRkJKViyDbN0HpPxJKk5vMDscQTI4QERGRU5J7DZK0BKjfBMqNiWaHQ0RERD4me36BLPsSSpuOULr3NzscwzA5QkRERE7Jph+Bi+d41ggREVEQkpPHYf38HaBmHSgjH4WiKGaHZBgmR4iIiMghyc2FrFoM1G0I1G9idjhERETkQ5KdWXAB1lLhsDzyLJRS4WaHZCgmR4iIiMgh2bIWOJ8BS6/BJfqbIiIiIrIn+fmw/ns6cPYMLA8/A6VCnNkhGY7JESIiIrqO5OVBVv4HuKEe0KiF2eEQERGRD8miWcCBPVCGPwylbkOzw/EJJkeIiIjoOrJ1HXD2NM8aISIiCjLWn36ArF0OpWsfWG67w+xwfIbJESIiIrJTcNbIIiChLtC0ldnhEBERkY/IkRTIl58Cic2hDBxjdjg+xeQIERER2ZFtG4Az6bD05lkjREREwULOnoH14zeA2IqwPDgFSkiI2SH5FJMjREREpJL8fMj3XwM1awPNbjY7HCIiIvIBybkK64xXgbxcWB59HkqZKLND8jkmR4iIiEj1v7NGhvCsESIioiAgViuss94HTvwBywOToFStYXZIpmByhIiIiAAUnjXyDVD9BqB5G7PDISIiIh+QFQuBnZuhDBwFpUnwXmuMyREiIiICAMivG4HTJ2G5i9caISIiCgayfRNk+UIo7W6Hckc/s8MxFZMjREREBLH+96yR+ASg+S1mh0NEREQGkz9SYZ39PlCnAZThjwT9FyNMjhARERHk101A+l+w3DUEioXDAyIiopJMLp6HdcZrQFQ0LI88AyUszOyQTMfRDxERUZATaz5kxdcFZ420aGt2OERERGQgyb1WkBjJzoRl3PNQosubHZJfYHKEiIgoyBWcNXKCZ40QERGVcCICmfMhcOwwLPc9AaVmbbND8hscAREREQWxgrNGFvKsESIioiAgKxdBftkApd9wKC3bmR2OX2FyhIiIKIjJLxv/e62Re3jWCBERUQkmO7dAls6HcnNHKHf+w+xw/A5HQUREREFK8vMhK74BqtcCWvAONURERCWVHE+Fdea7wA31oIweH/R3pnGEyREiIqIgJb9sBE7xDjVEREQlmXpnmjJlYRn3HJSwUmaH5Jc4EiIiIgpCBWeNfF1w1khznjVCRERUEsm1nILESFYmLI8+ByWGd6ZxhskRIiKiICTbNgCnT/JaI0RERCWUiEDmFt6Z5kkoNeuYHZJf42iIiIgoyEh+PuT7r4EaN/BaI0RERCWUfP815JeNUPqPhNKSd6QrDpMjREREQUa2rgdO/11w1ggvyEZERFTiWH/dBFn2FZS2naH0GGB2OAGByREiIqIgInl5kBULgZq1geZtzA6HiIiIdCbHfofMfh+o2xDKiEf5RYhGTI4QEREFEdmSDGScgqXPMA6WiIiIShg5l1FwAdbocrA8/AyUsDCzQwoYoWYHQERERL4hebkFd6i5oR7QtJXZ4RAREZGO5OoVWD96Bci5AssT06FElzM7pIDCM0eIiIiChGxaA5w7A0ufoTxrhIiIqAQRqxXWme8CJ/6A5cEpUOITzA4p4DA5QkREFAQk9xpk5SKgTgOgUQuzwyEiIiIdybfzgN3boAy+H0qTm8wOJyAxOUJERBQE5KcfgPMZsPTltUaIiIhKEuumHyGrv4XS6U4oXXqZHU7AYnKEiIiohJNrOZCV/wHqNQIaNDU7HCIiItKJHNoLmf8JkNgCypAH+AWIF5gcISIiKuFkYxJw8RzvUENERFSCSPpfsH78BlCpKixjJ0MJCTE7pIDG5AgREVEJJjlXC84aadAUSv3GZodDREREOpCsS7B++DIQEgLL+H9CKR1ldkgBj8kRIiKiEkySvwcyL8LSd5jZoRAREZEOJC8X1k/eAM5lwPLIs1AqVjE7pBKByREiIqISStGkhrgAACAASURBVC5nQ5IWA01aQanb0OxwiIiIyEsiApk3Azi8H8rox3h81xGTI0RERCWUrPkOuJwFS9+hZodCREREOpCViyBbkqH0GQpLm45mh1OiMDlCRERUAkl2JmTNMqBlWygJdc0Oh4iIiLxk/fUnyNL5UNp0hNJ7sNnhlDhMjhAREZVAsnoJcPUKLH141ggREVGgk9SDkFnvA3UToYx6jHefMwCTI0RERCWMXLoAWbscSuv2UOITzA6HiIiIvCBn0mGd8RpQPrbgAqxhYWaHVCIxOUJERFTCyKrFQG4ulLvuMTsUIiIi8oJkZ8H6wctAfj4sj02FUjba7JBKLCZHiIiIShA5fxayfiWUdp2hVIk3OxwiIiLykOTlwvrpm8CZ9IIzRnhcNxSTI0RERCWIfP81IAKlFy/URkREFKhEBDL/E+Dgb1BGPgqlfmOzQyrxmBwhIiIqIeRMOmTTj1Dad4NSsYrZ4RAREZGHZOUiyM9roPQeAku7LmaHExSYHCEiIioh5LsFQEgIlF6DzA6FiIiIPGTdtqHglr23dILSh9cP8xUmR4iIiEoA+es4ZNt6KJ17QSlXwexwiIiIyAPyewpkzr+Aeo2gjBzPW/b6EJMjREREJYD1uy+B8AgoPQaYHQoRERF5QE6dLLhlb2xl3rLXBEyOEBERBTg59juwcwuUbndDieIt/oiIiAKNZF6C9YOXAEWB5bEXoJQpa3ZIQYfJESIiogBnXTofiCoLpWsfs0MhIiIiN0nuNVhnvAqcy4Bl3HNQKlU1O6SgxOQIERFRAJND+4CUXVB6DoQSWdrscIiIiMgNYrVCZr0PpB6E5f4nodRtaHZIQYvJESIiogAlIrAu/QIoVwFKpzvNDoeIiIjcJN/Og2zfBGXgGCg33Wp2OEGNyREiIqJAtXc7cOQAlF6DoZQKNzsaIiIicoN1/SrI6m+hdOoJpVs/s8MJekyOEBERBSCxWmFd8gVQsQqU2+4wOxwiIiJyg+zdDvnq30CTVlCGPMhb9voBJkeIiIgCkPz6E3AiDUrfYVBCQ80Oh4iIiDSSP1Jh/fd0oEYtWB6cDCUkxOyQCEyOEBERBRzJy4Us+xKofgOU1u3NDoeIiIg0krOnYf3wZaBMWVjGvwAlItLskOi/mBwhIiIKMLLpR+BMOiz9R0Cx8FBOREQUCCQ7C9Z/vQRcuwbL41OhlKtgdkhkgyMqIiKiACI5OZAVXwN1E4HGN5kdDhEREWkgubmwfvw6cPpvWMY9C6VaTbNDoiKYHCEiIgogkrwcuHgelv4jefE2IiKiACBWK2TOB8DhfVBGPwalfhOzQyIHmBwhIiIKEJKdBUlaXHBl+xsTzQ6HiIiINJClX0B+2QCl33BYbulkdjjkBJMjREREAUJWLwYuZ8Ny9wizQyEiIiINrOtXQlYthtKhO5Q7/2F2OOQCkyNEREQBQC6chaxdDuXmjlBq3GB2OERERFQM2b0N8tX/FZzxOfQh/hzWzzE5QkREFABk+UIg3wql3zCzQyEiIqJiyNFDsH72FlCzNixjp0AJCTE7JCoGkyNERER+TtJPQDb9CKVjDygVq5gdDhEREbkgp0/C+tGrQEwFWB77J5TwCLNDIg2YHCEiIvJz1qXzgbBwKL0GmR0KERERuSCZF2H910uAWGF5bCqU6PJmh0QaMTlCRETkx+TYYWDHZijd+kKJLmd2OEREROSE5FyF9cNXgPNnYRn3PJQq8WaHRG5gcoSIiMhPiQisi+cCZWOgdOtndjhERETkhOTnw/p/bwFpR2B5YBKUug3NDoncxOQIERGRv9q/Czi0F0qvwVAiSpsdDRERETkgIpCvPgV++xXK0AehtLjF7JDIA0yOEBER+SGxWmH9di4QVxlKx+5mh0NEREROyPffQDauhtJzICyd7jQ7HPIQkyNERER+SH79CfjzGJS+w6CEhpkdDhERETlg/XkNZNmXUNp2hnL3CLPDIS8wOUJERORnJDcXsuQLoMYNUG7uYHY4RERE5IDs3Q6Z9xGQ2BzKyEehKIrZIZEXmBwhIiLyM7JhJXD2NCwDR0Ox8FBNRETkb+TYYVg/nQZUvwGWh5/mWZ4lAEdcREREfkQuZ0FWfFPwLVRiC7PDISIioiLk1ElYP3gZiCkPy+Mv8KLpJQSTI0RERH5EkhYD2ZmwDBhldihERERUhFw6D+v7UwEAlsdfhBJd3uSISC9MjhAREfkJOZcBWbMcSpuOUGrWMTscIiIisiFXL8P6r5eBSxdgeewFKJWrmR0S6YjJESIiIj8h330FiBVKv+Fmh0JEREQ2JC8X1o/fAE4cg+Whp6DcUM/skEhnTI4QERH5AfnrOGRzMpROvaDEVTY7HCIiIvovsVohs/8FHNgDZdR4KE1amR0SGYDJESIiIj9g/XYuEBEJpdc/zA6FiIiI/ktEIItmQ37ZCKX/KFja3W52SGQQJkeIiIhMJof2Ar/9CqXnQChR0WaHQ0RERP8lPyyBrFkG5fa7oPTob3Y4ZKBQswMAgI8//hg7d+5ETEwM3nnnHQDAN998g7Vr1yI6umCQeM8996Bly5ZmhklERKQ7sVphXTQbqBAH5fbeZodDRERE/2Xdsg7ynzlQWt0GZdB9UBTF7JDIQH6RHOnUqRN69OiBGTNm2L3eq1cv9OnTx6SoiIiIjCe//gT8cQTKvU9AKRVudjhEREQEQPZuh8z5F9CgacEx2sIfXZR0frGGExMTERUVZXYYREREPiW51yBLvgBq1obSpqPZ4RAREREAST0I66fTgOq1YHnkWShhYWaHRD7gF2eOOLN69Wps3LgRtWvXxsiRI5lAISKiEkWSvwfOnoZl1Hh+I0VEROQH5O8/Yf3wFSCmPCyPT4USWdrskMhHFBERs4MAgNOnT2PatGnqNUcuXLigXm/k66+/xvnz5/HII484nHbNmjVYs2YNAODNN9/EtWvXDIkxNDQUeXl5hsyb7LGtfYdt7Vtsb9/x97a2Zl5CxsP/QFj9xij/z3fMDscrRrZ1qVKlPJ7WF+MDf9/OShq2t++wrX2Hbe07xbV1fsYpnHvmISAvF+Vf/xShVav7MLqSxV/HBq747Zkj5cqVUx/ffvvtmDZtmtOyXbt2RdeuXdXnGRkZhsQUFxdn2LzJHtvad9jWvsX29h1/b2vr1zMhl7OR12eoX8ephZFtXa1aNY+n9cX4wN+3s5KG7e07bGvfYVv7jqu2luxMWKc9DWRdgmXy67gQFgFwvXjMX8cGrvjtObznz59XH//yyy+oUaOGidEQERHpR86kQ9Z9D+XW26HEJ5gdDhERUVCTnKsFP6U58zcs456DUrOO2SGRCfzizJH3338fKSkpyMzMxEMPPYRBgwZh//79SEtLg6IoqFixIh588EGzwyQiItKFfDsPCAmB0neo2aEQEREFNcnLK7j46tHDsDw0BUqDpmaHRCbxi+TIhAkT/p+9Ow/3sc7/OP56386xHPtWKRUpISVp34QWFaVNipQWoYX2ZaZ1ptJMm4p2FCJkqVAirVIh0tBCIvsS2Y/D/f798f1laoos55zPd3k+rmuuq+HMmed8rzPO1+vc9/35w681btw4QAkAAAXLZ30jn/ixrHkrWbmKoXMAAMhYHseJ43q/niS75BrZYceGTkJASXtbDQAA6cbdFQ98SSpbQXbauaFzAADIWO4uH/iS/LMPZC3aKDrxtNBJCIxxBACAQuITP5F++FbWorWsWPHQOQAAZCwfOUg+9k1Zk+ayMy4InYMkwDgCAEAh8Lw8+ZCXparVZMdy6ygAAKHEH74tH9ZXdmRDWcsrZGahk5AEGEcAACgE/t5b0rLFii5oJ4uKhM4BACAj+aRP5H2fkQ4+XNausyzir8RI4CsBAIAC5qtXyUcMlOo2kNWpHzoHAICMlDvlc8UvPCrVqKXo6ttkWUlxPgmSBF8NAAAUMH+zv7RhvaLz24VOAQAgI/ns7/TLY3dJVaoquu4uWbFioZOQZLhyBACAAuSL5sk/fFt2wqmyvfYJnQMAQMbxhT8p7naforLlFXW+V5ZTKnQSkhBXjgAAUIDiQb2k7KKysy8KnQIAQMbxZYsVP3a3lJWlcvd208osrhjBn+PKEQAACohPnyJ99YXsjJayMuVD5wAAkFF81QrFj98tbdygqMt9ytpjr9BJSGKMIwAAFACPNyse+JJUaXfZyc1D5wAAkFF83RrFj98rrfxZ0fX3yKpWC52EJMc4AgBAAfCP35Xmz1F0/mWy7KKhcwAAyBiem6v4qX9IC39S1OlOWY1aoZOQAhhHAADIZ75urXxYP+mAOtJhx4bOAQAgY/imPMXPdpVmfavoqptkB9UPnYQUwTgCAEA+85GDpNW/KGp5hcwsdA4AABnB483ynk9IX0+SXdJJ1uC40ElIIYwjAADkI1+6SD72DdkxjWTVDgidAwBARnB3ed9n5F98JDu/naITTg2dhBTDOAIAQD6KX+8tRUVk57QNnQIAQEZwd/ng3vKPRsvOaKnotHNCJyEFMY4AAJBP/NuvpUnjZU3Pk5WvGDoHAICM4CMHyUcPlTU6Q9aidegcpCjGEQAA8oHHmxW/9oJUoZLsVH5iBQBAYYjHjZAP6ys7+iRZq/Y86ws7jXEEAIB84B+PkX6aLTu/naxYsdA5AACkvXj8WPmrz0n1jpRder0s4q+32Hl89QAAsIsSR/f2lfavIzv8+NA5AACkPZ80Xt77Kal2PUVX3yrLygqdhBTHOAIAwC7yEQOlNasUtbqSy3kBAChg/vVkxS88Iu1XU1GnO2XZRUMnIQ0wjgAAsAt88QL52DdlxzaW7bt/6BwAANKafz9d8TMPSnvurej6u2XFS4ROQppgHAEAYBfEg3pKWdkc3QsAQAHzOTMVP3W/VKGyoi73yXJKhU5CGmEcAQBgJ/n0L6Wpn8vObCkrWz50DgAAacvnz1H8+D1STilFN9wvK1MudBLSDOMIAAA7wTdtUjzgRanyHrKTm4fOAQAgbfniBYofv1vKylZ04z9kFSqHTkIaYhwBAGAn+PsjpIU/KWp5BQ+CAwCggPjyJYof+7u0ebOiG++X7VYldBLSFOMIAAA7yFf/In9jgFSnvlTvyNA5AACkJV/5s+LH7pLWr1d0w32yPfcJnYQ0xjgCAMAO8qF9pI0bOLoXAIAC4qtXJW6l+WWFos73yPapEToJaY5xBACAHeBzZsk/flfWqJmsyt6hcwAASDu+bo3iJ+6Wli5SdO3fZTVqhU5CBmAcAQBgO7m74gHPS6XKyJpfGDoHAIC04xvWKe52nzR/rqKOd8hqHRI6CRmCcQQAgO3kn38ozZwhO+cSWU6p0DkAAKQV35ir+OkHpB+/V9T+FtnBDUInIYMwjgAAsB18w3r54N7SvvvLjjs5dA4AAGnF8/IUP/OQ9N3XsstvkB12TOgkZBjGEQAAtoOPHCitXK7oovayiG+fAADkF9+0SfHz/5a+niy75BpFRzUMnYQMxLs7AAD+gi+aLx89XHZMYx4KBwBAPvLNm+UvPSZNmSC7qL2iE04NnYQMxTgCAMA2uLvi116QihaVnXdp6BwAANKGx7G895PyiR/LLminqHGz0EnIYIwjAABsy9TPE5f5Nr9IVrZ86BoAANKCu8v79pBPGCdr0UbRqeeETkKGYxwBAGArfGOu4tdelKrsLWt0ZugcAADSgrvLB7wg/2i07IyWis5sGToJYBwBAGBrfPRQadnixENYs7JC5wAAkPLcXT64l/y9t2SntpC1aB06CZDEOAIAwJ/y5UvkIwfLGhwnq10vdA4AACnP3eVD+8hHD5M1biY7v53MLHQWIIlxBACAPxW/9qJkkl1weegUAADSgr85QD5qsOzEprJWVzGMIKkwjgAA8D/860nSlxNkZ14oq1g5dA4AACkvHjFQ/mZ/2XFNZK07MIwg6TCOAADwG56Xp7j/89Lue8lOaRE6BwCAlBePHiof1ld2VENZ22tlEX8NRfLhqxIAgN/w0UOlJQsTD2HNzg6dAwBASovHvCEf1Et2+PGydl1kUZHQScCfYhwBAOD/+bLF8pEDpQbHyg6qHzoHAICUFo8bIX/tRemwY2VX3CgrwjCC5MU4AgDA/4tfe0mSKWp5RegUAABSWvz+KPmrz0mHHqXoqptkWVmhk4BtYhwBAECST5soTZkga3ahrAIPYQUAYGfFH42W93tGOvhwRe1vlWVxmyqSH+MIACDjed7GxENY99hLdsrZoXMAAEhZ8Sdj5X26S3UPU9Txdp7fhZTBOAIAyHj+9hBp6SJFF13NT7cAANhJ8fix8peflGrXU9TxDll20dBJwHZjHAEAZDRfskA+cpDsiBNkdQ4NnQMAQEqKPx0n7/2kVOsQRdf8TVa0WOgkYIcwjgAAMpa7K371OSkrS9by8tA5AACkpHjC+/Je3aQDD1Z0zd8ZRpCSGEcAAJlr8qfSf76Und1aVq5i6BoAAFJO/NkH8p5PSDUPUnTtXbJiDCNITYwjAICM5BvWKR7wglS1uqzRmaFzAABIOfHnH8pfelw6oI6i6xhGkNoYRwAAGcnffE1auVxRm46yIkVC5wAAkFLizz+Uv/iYdEBtRdffLStWPHQSsEsYRwAAGcfnz5GPGS474VRZjVqhcwAASCm/H0buYRhBWmAcAQBkFI9jxX17SDklZee2DZ0DAEBKYRhBumIcAQBkFP9kjDRzhuz8drJSZULnAACQMuLPPmAYQdrKCh0AAEBh8dW/yF9/Wap5kOzYJqFzAABIGfGE9xOn0hxQh2eMIC0xjgAAMoYP6iltWK+odUeZWegcAABSQvzpOHmvbonjeq+7i2EEaYnbagAAGcG/+Ur+6TjZaefI9twndA4AACkhHj9W3usJ6cC6iq7jihGkL64cAQCkPc/LU9zvGanyHrIzW4bOAQAgJcSfjJW//KRU6xBF1/xdVqxY6CSgwHDlCAAg7fk7r0uL5iu6+GpZUd7YAQDwV+KPRieGkdr1FF3LMIL0x5UjAIC05osXyEcMkh1+vKxug9A5AAAkvfiDt+V9e0h1D1PU8Q5+sICMwDgCAEhb7q64bw8pO1t24RWhcwAASHrxuBHyV5+TDj48MYxkZ4dOAgoFt9UAANKWfzpO+uYr2bltZeUqhs4BACCpxWOGJ4aRQ49iGEHG4coRAEBailetlA96SapRS3Zi09A5AAAktXj0UPmgXtJhxyi66mZZFsMIMgvjCAAgLa3u/bS0fp2iS66RRVwoCQDA1sQjBsqH9U08n+uKG2VZ/DURmYevegBA2vEZU7Vh3EjZ6efL9to3dA4AAEnJ3eVv9pe/OUB29EmyyzrLihQJnQUEwTgCAEgrnrdRcd9nVGSPveTNLgydAwBAUnJ3+dA+8lGDZcc1kbW9VhYxjCBzFeg40rFjRy1fvlySVL9+fd1xxx1b/dghQ4ZowIABKlWqlHr27FmQWQCANOYjBkpLFqj0fU9qNUcPAgDwB+4uH9xLPnqY7MSmstYduAUVGa/A/h+wevXqLcOIJE2bNk3r1q3b6sfPnj1bklS9evWCSgIApDmf96P87ddlRzdSsUMOD50DAEDS8TiW938+MYw0biZr05FhBFABjiM//PDDln8uVaqUNm3apIkTJ/7lxzOOAAB2hsebFb/ytFSipKzlFaFzAABIOh5vlvftIR83QnbqObJWV8nMQmcBSaHAx5HSpUurUaNGkqTPPvvsTz92zZo1Wrp0qSRpv/32K6gkAEAa83GjpNnfyS68Ula6TOgcAACSim/eLO/VTf7RaFmzC2XnX8YwAvxGgY0jv94ms99+++moo46SJE2ZMkUbNmz4w8f+9ioTxhEAwI7y5UvlQ1+R6h4mO6ph6BwAAJKKb9okf+ER+YT3ZS3aKDq7NcMI8D8K/MqR6tWr64ADDlCFChWUl5enyZMn/+Fjfx1SSpQood13372gkgAAacjdFfd7RpIUtenEmz0AAH7D8zYqfrarfNInspZXKDqzZegkICkVyDiyZs0aLVmyRFJiHDEzHXnkkZKkCRMm/OHjfx1S9ttvP97UAgB2iH/+oTRtoqxFa1nF3ULnAACQNDw3V/HT/5Smfi67uIOiU84OnQQkrQIZR369EkT67wNWf7215ssvv9TGjRv/9ON5GCsAYEf46lXyAS9I1Q6QNW4WOgcAgKTh69cp7naPNOMr2WWdFTU6I3QSkNQKdBwpWbLklttkateurTJlyig3N1dTpkzZ8rHr1q3T4sWLJfG8EQDAjvGBL0nr1yq69FpZVCR0DgAAScHXrlb8+N3SD9/KrrpJ0XFNQicBSa9AxpFfb5OpVq3alttkoijS4YcfLun3t9bMnj1b7i6JK0cAANvPp02STxgna3qerCrfPwAAkCRftVLxI3+XfvpBUYfbFR1xQugkICUU6JUj/zt2HH300ZKkSZMmadOmTZL+O6QUL15cVapUKYgcAECa8Q3rFPftLlXZW3bmhaFzAABICr5iueJ/3yktma/o2rtkhx4VOglIGfk+jqxbt06LFi2S9MfbZOrWraucnBytX79eX331laT/DinVqlVTFBXY4TkAgDTir78irViu6NLrZNnZoXMAAAjOly5S/K/bpZXLFXW+T3ZQ/dBJQErJ9zXixx9/3OptMllZWWrQoIGk/95a89sjfwEA+Cv+3X/k74+UNWkuq1ErdA4AAMH5wp8Sw8j6dYpu/Kes5kGhk4CUk+/jyF/dJvPrqTUTJ07UmjVrtHDhQkk8jBUA8Nd8Y67il5+SKu0ua9EmdA4AAMH53FmK/3WHFMeKbnlQVv2A0ElASiqwcWRrt8kceuihKlasmNasWaMRI0bwMFYAwHbzNwdISxYoanutrFjx0DkAAATlM2ckHr5atKiiW7vK9to3dBKQsvJ9HNnaw1h/VbRoUdWvn7j/bcSIEVt+rWrVqvmdAgBIIz5npnz0UNnxp8hq1wudAwBAUD59SuK43tJlEsPI7nuGTgJSWr6OIxs2bNCCBQskbftKkF9vrdmwYYMkad999+VhrACArfJNeYp7dZPKlJNd0C50DgAAQfmXExQ/db+0WxVFt3WVVdwtdBKQ8vJ1kfjtw1i39QyRww47TNm/OV2AW2oAANviIwZK8+couuQaWU6p0DkAAAQTfzpO8bNdpX1qKLr5QVmZ8qGTgLSQlZ+frFatWho4cOBfflyJEiXUr1+//PyvBgCkKZ87Sz5qsOzoRrJDjgidAwBAMPG4kfJXn5VqHaLomr/JipcInQSkjaS5l6VHjx668sorddNNN/3h99544w21bNlSq1atClAGAAglcTvNk1KpMrJWV4bOAQAgCHdX/NZriWGk3pGKrr+bYQTIZ0kzjpx00km68847//Dry5Yt07Rp01SpUqUAVQCAkHzU69K82YradJSVLB06BwCAQufu8kE95cP7yY5upKjjHbLsoqGzgLSTNONInTp1VKrUH+8jf/nll9W6dWuZWYAqAEAoPm+2fMRA2ZEnyg49OnQOAACFzjdvlr/8lPzd4bLGzWTtOsuKFAmdBaSlfH3mSH6bOHGiKlSooGrVqoVOAQAUIt+0KXE7TU5JWav2oXMAACh0npen+MVHpMmfypq1kp11ET8wBgpQ0o4jubm5GjJkiP7+97//5ceOGTNGY8aMkSR17dq1wG7BycrK4vaeQsJrXXh4rQsXr/f2WTOwl9bOnaWytz6g4tW3fvrZtvBaF55kfa0L4/1Bsv5vT1e83oWH17rw/NlrHa9fp18evkMbp36hUpd3VsnmFwaqSy98XReeVHytzX89ezcJLFmyRA8//LAeffRRzZ07V/fff7+KFSsmSVq+fLnKly+vhx56SOXKldvm51mwYEGB9FWqVEnLli0rkM+N3+O1Ljy81oWL1/uv+dwfFD94k6zBcYquunmnPw+vdeEpyNd6zz33zLfPVRDvD/g6K1y83oWH17rw/O9r7WtWKX7yfmnOTNml1yk6tknAuvTC13XhSZX3Br+VtFeO7LPPPnrxxRe3/PtrrrlGDz30kMqUKROwCgBQkBKn03RLnE5zEbfTAAAyi/+8TPET90hLFynqeDvP3AIKUdKMI0888YSmT5+u1atXq0OHDmrZsqUaN24cOgsAUIh8xKDE6TTX/E1WijEcAJA5fNH8xDCydrWiLvfKDjw4dBKQUZJmHOnSpcs2f7979+6FVAIACMHnzJKPHCg7ppHs0KNC5wAAUGh8zizF3e6V3BXd/IBs3/1DJwEZJ2nGEQBA5vK8PMW9npDKlJNdeFXoHAAACs3GaZMUP3KnlFNK0Q33yfaoGjoJyEhR6AAAAPzNV6X5cxS1vVZWslToHAAACoVPHq8V998ola+k6PZ/MYwAAXHlCAAgKJ85Q/72UNkJp8oOPjx0DgAAhSL+8B1532eUXbOONne8Q1aydOgkIKMxjgAAgvHcDYp7Pi5VqCRreXnoHAAACpy7y0cOkg/rK9VtoPJ/+5eWr1kbOgvIeIwjAIBgfHBvadliRTc9ICueEzoHAIAC5XEsf+1F+XtvyY5qKLuss6x4CYlxBAiOcQQAEIRP/1L+/kjZyWfLDqwbOgcAgALleXnyXk/Iv/go8b3vgnayiEdAAsmCcQQAUOh83RrFvZ6UquwtO6dN6BwAAAqUb1inuMdD0oypsvMulZ12rswsdBaA32AcAQAUOu//grRqhaJr7pQVLRY6BwCAAuOrVip+8n7ppx9kl3VWdFyT0EkA/gTjCACgUPmkT+QTxsmatZJVOyB0DgAABcaXLFT8xD3SLz8r6vQ3Wb0jQicB2ArGEQBAofGVPyvu00OqdoDszJahcwAAKDA+Z5bibvdKcazoxn/KatQKnQRgGxhHAACFwt0Vv/yklJer6IobZFl8CwIApCef/qXiHl2lUqUVdb5XVqVq6CQAf4F3pgCAQuEfjJK+niy7uINsD94kAgDSU/zZB/Je3aQ99lLU5V5ZuYqhQJs3rwAAIABJREFUkwBsB8YRAECB80Xz5YN6SgfVl510eugcAADynbvLRw+TD+4l1aybeOh4TqnQWQC2E+MIAKBA+aZNil96TMoupuiy6zm6EACQdjyO5QNfko99U3b48bLLb5BlZ4fOArADGEcAAAXKRw6UfvxeUYfbuLQYAJB2PG+j/KXH5ZM+kZ18luyCy2VRFDoLwA5iHAEAFBifOUP+1kDZMY1kDY4LnQMAQL7ytWsU93hA+u4/sgsuV3Rqi9BJAHYS4wgAoED4+nWJ22kqVpZddHXoHAAA8pUvX5o4qnfJQtlVNys68sTQSQB2AeMIAKBAeP/npOVLFd36oKxETugcAADyjc/9QfGT90sbcxMn0tQ6JHQSgF3EOAIAyHfxFx/JPx0na9ZKtn+d0DkAAOQbn/6l4me6SiVKKrqtq2yvfUMnAcgHjCMAgHzlPy+V9+0hVa8pa3Zh6BwAAPJNPP49+StPSVX2VnT9PbLyPGgcSBeMIwCAfOPxZsU9n5A2b1Z05Y2yIkVCJwEAsMvcXf7Wa/I3XpVq11PU4XZZTsnQWQDyEeMIACDf+NtDpG+nyS69TrbbnqFzAADYZb5pk7xvd/knYxOnr7W9VpaVHToLQD5jHAEA5Av/4Vv58H6yBsfJjjs5dA4AALvM169T/GxXafqUxHO0zrpIZhY6C0ABYBwBAOwyX79O8YuPSuUqyi65hjeOAICU5yuWJ06kWTBHdul1io4/JXQSgALEOAIA2GX+6nPSsiWKbnlQVrJU6BwAAHaJ/zQ7MYysX6fourtldQ8LnQSggDGOAAB2STzhffmEcbLmrWQHcGwvACC1+deTFD/7L6lETuKo3r2rh04CUAgYRwAAO82XLpL3e0bav7bsTI7tBQCktvjDt+X9npX22jdxxQhH9QIZg3EEALBTfNMmxS88Ilmk6MqbOLYXAJCyPI7lQ/vI335dqttA0dW3yIrnhM4CUIgYRwAAO8WH9ZVmf6fo6ltlFXcLnQMAwE7xjbmKez4uTRova9hUdtHVDP5ABmIcAQDsMP96svydIbITm8oOPz50DgAAO8VXrVD89APSj9/LLmgnO6UFJ64BGYpxBACwQ3zlz4mfsO21r+zCK0LnAACwU3zB3MSJNKtXKupwu+ywY0InAQiIcQQAsN08jhPDSO56RVc9ICtaLHQSAAA7zKdPUfzsw1LRoopufkhW/YDQSQACYxwBAGw3f/t1acZUWdtrZXvtEzoHAIAdtuVEmip7J06kqVg5dBKAJMA4AgDYLj5zhnx4P9kRJ8iOPyV0DgAAO8TjzfLXX5aPHibVPUxR+1tlJTiRBkAC4wgA4C/5mlWKX/i3VHE3WZtOPKwOAJBSPHeD4hcflaZ8Jmt0puzCKzmRBsDvMI4AALYp8ZyRJ6RVKxXd/i9ZTsnQSQAAbDdfsVzx0/+QfvpR1qq9oibNQicBSEKMIwCAbfJ3h0vTJsouai/bd//QOQAAbDefM1Px0/+U1q9XdN3fZQcfHjoJQJJiHAEAbJXP+kY+9BXpsGNkjc4MnQMAwHbzSZ8kTlgrXU7R7Q/LqlYLnQQgiTGOAAD+lK9drfj5f0vlKym69DqeMwIASAnuLh85SD6sr1SjlqJOd8rKlAudBSDJMY4AAP7A3RX36ib9siLx07acUqGTAAD4S563Uf7yU/LPPpAd1VB26XWy7KKhswCkAMYRAMAf+Oih0tTPE0/zr3ZA6BwAAP6S/7JCcY8HpR++lZ3dWnZmS656BLDdGEcAAL/j330tH/KK1OBYWZPmoXMAAPhLPneW4u4PSGtWK+pwu6zBsaGTAKQYxhEAwBa+aoXi5x+RKu2h6NLr+YkbACDp+eTxil96XCpZWtFtXWX71AidBCAFMY4AACRJHm9W/MKj0ro1irrcIyuREzoJAICt+t2DV6vXTDx4tVyF0FkAUhTjCABAkuRv9Je++Up22fWyqtVD5wAAsFWemyt/+Un5Fx/x4FUA+YJxBAAg/3qSfMRA2XFNFB13cugcAAC2yn9elnjw6txZsnMvlTU9l9tAAewyxhEAyHC+bLHiFx+T9tpXdlGH0DkAAGyVz/pG8TMPSRs2KLrmb7J6R4ZOApAmGEcAIIN53kbFzz4sxbGiTnfIihULnQQAwJ+Kx78n79NdKl9R0Q3/kO21T+gkAGmEcQQAMpj3f16aMzPx07fd9gydAwDAH/jmzfLXe8vfHS4deLCiDrfJSpUJnQUgzTCOAECGij9+V/7RaNnp58sOPSp0DgAAf+Br1yh+/t/S9C9ljZvJLrhclsVfYQDkP/5kAYAM5HNmyfs9K9WuJ2vROnQOAAB/4At/Uvz0P6XlS2Vtr1V0wqmhkwCkMcYRAMgwvnZ14mF2pcsquupmWVQkdBIAAL/jUz5T/NJjUtFiim5+QLZ/7dBJANIc4wgAZBCPNyt+4RFp5c+Kbn1IVrps6CQAALbwOJaPGCh/41Vp3/0VdbpTVqFS6CwAGYBxBAAyiA9/VfrPl7JLOsn2OzB0DgAAW/iGdYp7PiF9OUF2TCNZm06yopyiBqBwMI4AQIbwyZ/KRw6SnXCqohObhs4BAGALX7JAcfcHpUXzZBdeKWvSXGYWOgtABmEcAYAM4At/Svw0rnpN2UVXh84BAGALnzZJ8YuPSBYp6nKfrHa90EkAMhDjCACkOV+3NvHTuKJFFXW4XZadHToJAAC5u3zkIPnwflLVaonni1TaPXQWgAzFOAIAaczjWHHPx6WlCxXd+E8eagcASAq+YZ3iXt2kyZ/Kjmwoa3utrBjPFwEQDuMIAKQxf3OANPVzWaurZAfWDZ0DAIB80TzFPR6SFs+XtbxCdvJZPF8EQHCMIwCQpnzyePlbA2THNpE1bhY6BwAA+ZQJiWdgFcni+SIAkgrjCACkIZ/3438fwNqmIz+RAwAE5fFm+Rv95SMGSvvur6jjHbKKlUNnAcAWjCMAkGZ87WrFPR6Uiuco6nSHLLto6CQAQAbztWsSp9F8PVl2XBNZ6458bwKQdBhHACCN+ObNip/7l7RimaKbH5SVqxg6CQCQwXzuLMXPdJVWLJe16SQ78TSuZgSQlBhHACCN+OBe0oypskuvk9WoFToHAJDB4vFj5X2fkUqWVnTLg3xfApDUGEcAIE3EH42Wj3lD1qS5ouNPCZ0DAMhQnpcnH/ii/P1R0oEHK2p/i6xMudBZALBNjCMAkAb8++nyfs9KdQ6VXXB56BwAQIbyn5cqfvZhafZ3stPOlZ1ziaxIkdBZAPCXGEcAIMX5ssWKn3lIqrS7ova38iYUABCET5+i+IVHpLw8RR1ulzU4NnQSAGw3xhEASGG+Yb3i7g9ImzYpuvZvspKlQicBADKMx7F81GD58H7SHlUTJ6XtUTV0FgDsEMYRAEhRHseKX3pcmj9XUed7eCMKACh0vnaN4pcek6ZNlB3ZUHZJJ1nxEqGzAGCHMY4AQIryYX2kKRNkF14pO6h+6BwAQIbxOTMTx/Su/Fl2cQfZSadzTC+AlMU4AgApKP5krHzU67KGTWVNmofOAQBkEHeXf/C2/LUXpDLlOKYXQFpgHAGAFOPffS3v012qXU/Wqj0/pQMAFBrfsF7et4f8sw+kuocpuuJGWakyobMAYJcxjgBACvElCxT3eEiqvIeiDrfJsvhjHABQOHzhT4nbaBbNk53dWnbGBbIoCp0FAPmCd9UAkCJ87RrFT/1DMim67i5ZDifTAAAKRzxhnLxPD6lYcUVd7pPVOTR0EgDkK8YRAEgBvilP8bNdpaWLFd14v2y3KqGTAAAZwDfmyge8IP9otFTzIEVX3SwrVzF0FgDkO8YRAEhy7i7v20P65itZuy6ymnVDJwEAMoAvXqD42YelebNlp5+fuJWmSJHQWQBQIBhHACDJ+chB8k/Gypq3UnRs49A5AIAMsOHjMYq7PyQVyVJ0/d2ygw8PnQQABYpxBACSWPzZB/JhfWVHnyRrflHoHABAmvONufLXXtIvH74t7Xegova3yipWDp0FAAWOcQQAkpR/9x95725SzYNkba/jyF4AQIHyRfMVP/cvad5s5bRorQ2nncepaAAyBn/aAUAS8kXzFfd4UKq0u6JOd8qys0MnAQDSWPzZB4nTaLKyFF13l0o3Pl25y5aFzgKAQsM4AgBJxletUNztXimKFF13t6xk6dBJAIA05bm58gHPyz9+V6pRS1H7W2QVuI0GQOZhHAGAJOK5GxQ/+Q9p1QpFNz/Ikb0AgALj8+cqfu5hadG8xGk0Z13MbTQAMhZ/+gFAkvDNmxP3es/9QdE1d8qq1wydBABIQ+4u//hd+YDnpWIlFHW5V1anfugsAAiKcQQAkoC7y199Vpo2Uda6o6zekaGTAABpyNetlfftIf/iI6l2PUVX3CgrWz50FgAExzgCAEnARw6Sf/iO7PTzFJ10eugcAEAa8h++VfzCI9LPS2Ut2iRupYmi0FkAkBQYRwAgsPiTMfJhfWVHNpS1uCR0DgAgzXgcy98ZKh/eVypXUdEtD8n2rx06CwCSStKMIz169NDkyZNVtmxZPfroo5KkAQMGaOLEiTIzlS1bVp06dVKFChUClwJA/vGpX8hfeVqqc6is3fX8BA8AkK/8lxWKez4uTZ8iNThWUdtrZTmlQmcBQNJJmnHkpJNOUtOmTdW9e/ctv3bWWWepVatWkqSRI0dq8ODBat++fahEAMhXPusbxc8/LO29n6KOt8uyskMnAQDSiE+bqLhXNyl3vaxNJ9mJp8nMQmcBQFJKmnGkTp06WrJkye9+LScnZ8s/5+bm8oc5gLThC+cpfuofUtkKiq6/W1Y856//QwAAbAfPy5O/3ls+9k1pr30VtX9Atuc+obMAIKklzTiyNf3799eHH36onJwc3XPPPaFzAGCX+Yrlip+4R4oiRV3uk5UpFzoJAJAmfOE8xc//W5o3W9a4mez8y2TZRUNnAUDSM3f30BG/WrJkiR5++OEtzxz5raFDhyovL08tW7b8w++NGTNGY8aMkSR17dpVGzduLJC+rKwsbdq0qUA+N36P17rw8FoXrmj9Wi25rb3ipYtV/p/dlV3jwNBJaYuv7cJTkK910aI7/5e6wnh/wNdZ4eL13jp31/rRw7W6ZzdZ8RIqe+3fVOyI43b68/FaFx5e68LDa114kvW9wbakzDiydOlSde3a9U9/738tWLCgIPJUqVIlLVu2rEA+N36P17rw8FoXHt+wXkWeul95P3ynqPM9slqHhE5Ka3xtF56CfK333HPPfPtcBfH+gK+zwsXr/ed89SrFrzwlTflMqnOoonZdZOV27RADXuvCw2tdeHitC0+qvDf4raS+rWbhwoWqUqWKJGnixIkF9iIAQEHzTXmKn+mqeOYMRR1uZxgBAOQLnz5Fcc8npLWrZC2vkDVpzslnALATkmYceeKJJzR9+nStXr1aHTp0UMuWLTV58mQtXLhQZqZKlSpxUg2AlOTxZvlLj0vTv1SZa+/U2npHh04CAKQ4z9soH9pH/u5wqcreiYd777Nf6CwASFlJM4506dLlD7/WuHHjACUAkH/cXf7qc/KJH8vOb6cSTZppLZdzAgB2gc+fo/iFR6T5c2QnnSE7v52sWLHQWQCQ0pJmHAGAdORDX5F/8Las6XmKTjsndA4AIIV5HMvfe0v++stSiRxF190lO+SI0FkAkBYYRwCggMQjB8lHvS5r2FR2btvQOQCAFOYrlivu/aQ0/UvpkCMUXXodR8EDQD5iHAGAAhC/95Z8aB/ZUQ1lF3eQmYVOAgCkKJ/4seI+PaRNebLWHROjO99XACBfMY4AQD6Lx4+V939eOvQo2WWdOTUAALBTfN1aef/n5BPel6rXVHT5DbI99gqdBQBpiXEEAPKRTxov7/2UVLueova3yLL4YxYAsOP822mJI3pXLpc1v0h2xgV8TwGAAsSfsACQT3zqF4nTA/arqajTnbLsoqGTAAApxvM2yof0kY8ZLu22p6LbHpbtd2DoLABIe4wjAJAPfPqXip99SKpaTdH198iKlwidBABIMT5npuKXHpcW/iRrdIbsvMtkxYqHzgKAjMA4AgC7yL/9WnH3B6Q9qiq64T5ZTsnQSQCAFOKbN8tHDZK/9ZpUupyiLvfJDqofOgsAMgrjCADsAp/1jeKn7pcq7q7oxn/ISpYOnQQASCG+8KfEs0V+/F52ZEPZxVfLSpYKnQUAGYdxBAB2kv/4veJu90plyyeGkdJlQycBAFKEx7F8zBvyoX2k4sUVdbhN1uC40FkAkLEYRwBgJ/icmYofv1sqWVrRTf+UlasQOgkAkCJ86SLFvbtJ3/1HqnekorbXyMqUD50FABmNcQQAdpDPmaX4sbulEiUV3fyArELl0EkAgBTgcSz/4G35672lKJK16yw7prHMLHQaAGQ8xhEA2AE+d5bix+6SSuQkhpGKu4VOAgCkAF++RPHLT0kzpkp16iu69FrGdQBIIowjALCdfO4PiStGipdI3EpTaffQSQCAJOfu8o9Gywf2lCTZJdfITjiVq0UAIMkwjgDAdkhcMXK3VKxY4oqRynuETgIAJDlfvkTxK09L06dItQ5RdNn1XHEIAEmKcQQA/oL/+H3i4avFcxJXjDCMAAC2wd3lH74jH9RLkmStO8pOPE0WRYHLAABbwzgCANvgs79T/Pg9Us7/P3yVW2kAANvgyxYnni3yzVdS7XqK2l7L9w4ASAGMIwCwFT7rG8Xd7pVKlVF00wOyijw4DwDw5zyO5e+PlA95RZLJLukkO+E0ni0CACmCcQQA/oR/P11xt/uksuUSw0iFSqGTAABJyhcvUPzyk9L30xMn0bS9hmeLAECKYRwBgP/h06co7v5PqULlxDNGylUMnQQASEIeb5aPeUM+rJ+UnS27rLPs2MZcLQIAKYhxBAB+w6d+ofjZrtLueyq68X5ZmfKhkwAAScjn/Zh4tsiP30uHHqWodUdZuQqhswAAO4lxBAD+n0/8WPGLj0pVqyu64T5ZydKhkwAAScbz8uQjB8pHDZZySsna3yI7/HiuFgGAFMc4AgCS4vHvyXs/KdWopei6u2Q5JUMnAQCSjM+cofiVp6WFP8mOaSRreYWsVJnQWQCAfMA4AiDjxe+9Je//fOLIxWv+JitWPHQSACCJ+IZ18iF95O+PlMpXUtT5HlndBqGzAAD5iHEEQMZyd/mI1+TDX03cL97+Fll20dBZAIAk4lO/UNzvGWnlclmjM2XntJEVzwmdBQDIZ4wjADKSu8sH9pSPGZ64NPrS62VFioTOAgAkCV+1Qj7gRfkXH0l77qPo6ltlNWqFzgIAFBDGEQAZxzdvlvd5Wv7JWFnjZrILr5RFUegsAEAS8DiWfzJGPriXtDFXdnZrWdNzZVnZodMAAAWIcQRARvG8jYpfeET6coKseStZ84s4YQAAIEnyhT8p7tNd+n66VLOuojadZFWqhs4CABQCxhEAGcPXrVHc/QHpu//ILrxS0clnhU4CACQBz9soHzVYPnKwVKy47NLrZMedzHgOABmEcQRARvCVPyvudq+0cJ7sypsUHdUwdBIAIAn4jKmK+z0rLZ4vO7Kh7MIrZGXKhc4CABQyxhEAac8XL1D8+N3SmlWKrr9LVqd+6CQAQGC+aqV8UE/5hPelynso6nKf7CC+PwBApmIcAZDWfPb3ip+6X3JXdPMDsmoHhE4CAAT03weu9pZyN8jOaCk78wJZ0WKh0wAAATGOAEhbPvULxc//SypTTlHne2V77BU6CQAQkM+brbjvM9Ksb6QD6ii65BpZlb1DZwEAkgDjCIC0FH/wtrzfs9I++yVupSlTPnQSACAQ37BO/kZ/+dg3pZxSsss6y45tzANXAQBbMI4ASCvuLh/WTz5yoHTw4Yra3yIrXiJ0FgAgAHeXJn+q+LUXpRXLZCecKju3raxUmdBpAIAkwzgCIG34pjz5K0/LPx2XeAPcuqOsSJHQWQCAAHzxAsWvPidN/1KqWl3R1bfKatQKnQUASFKMIwDSgq9drbjHQ9J3X8vObi07syWXSwNABvLcXPmoQfJ3hkjZRWWtrpKddAZjOQBgmxhHAKQ8X7JA8ZP/kJYvll15k6KjGoZOAgAUMneXpn6ueMAL0vIlsqMays5vJytXIXQaACAFMI4ASGk+c7ri7g9IkqIb/yk7oE7gIgBAYfMlCxT3f0H6epJUZe/E0e0HHhw6CwCQQhhHAKSs+LMP5L27SRV2U9T5btlue4ZOAgAUIs/NlY8cJB89RMrKlrW8QtboTFkWb3EBADuG7xwAUo7HsXx4P/nIQVLNuoo63s7JAwCQQdxdPvET+eCe0s/LZEc3kp1/mawsx7YDAHYO4wiAlOIb1ivu+bj05YTEiTQXXy3Lyg6dBQAoJD7vx8RzRb6dJu1dXdGVN3NLJQBglzGOAEgZvnyp4qf/Kc2fI7vwSlmT5pxIAwAZwteu0aphrygeNUQqUTJxXPuJp8oiTqEBAOw6xhEAKcFnTlf8TFcpb6Oi6++S1W0QOgkAUAg83iz/cLR8eF+tX7dWdmJT2dkXczslACBfMY4ASHrxB2/L+z8vVaycOIGgyt6hkwAAhcC/nZa4hWbej1LNuqrQ4Rb9UprnigAA8h/jCICk5Zvy5P2fl3/4jlS3gaKrbpLllAqdBQAoYL50keLXe0uTxksVd1PU4TbpsGOVXbmytGxZ6DwAQBpiHAGQlPyXFYqf7SrNnCE7/TxZizbcVw4Aac43rEsczfvucCkqIjvrYtlp58iKFgudBgBIc4wjAJKOz5yh+LmHpXVrZe1vVXTE8aGTAAAFyOPN8vHvyYf2kVatlB3TSHZOW1n5iqHTAAAZgnEEQNJwd/m4EfKBL0kVKiu6/R7Z3tVDZwEACpB/85XigS9JP82WatRSdO3fZdVrhs4CAGQYxhEAScFzN8j7dJd/9oF0yBGKLr9BVpLniwBAuvJF8xPPFZnymVShsuyqm2VHnMAR7QCAIBhHAATnixcofuYhacFc2dmtZWdcIIui0FkAgALga1bJRwyUjxshZReVndtW1qQ5zxUBAATFOAIgqPiLj+WvPCUVyVJ0/d2yug1CJwEACoDnbZS/N0I+cqC0fr3s+JNlLVrLynA0LwAgPMYRAEF4Xp58UM/ETw73O1BR+1tlFSuHzgIA5DN3l3/xkXzIK9LyJYmj2c+/TLbXvqHTAADYgnEEQKHzpYsUP/cvac5M2SlnJy6pzsoOnQUAyGf+7deJ54rM/k6qWl3RDffJ6tQPnQUAwB8wjgAoVD7xY8WvdJckRZ3ulNU/OnARACC/+fy5ioe8LH31hVS+kuyyzrJjTpJFRUKnAQDwpxhHABQKz82VD3xR/uE7UvWaiq66WVZ5j9BZAIB85CuWy9/sL/94jFS8uOzcS2VNmvGwVQBA0mMcAVDgfP4cxc//O3EaTdPzEifSZPHHDwCkC1+3Rv726/Kxb0qb48QgckZLWekyodMAANgu/O0EQIFxd/kHo+QDe0olcrjXHADSzH9PoBkkrV8rO6qh7KyLuTIQAJByGEcAFAhftUJx76ekaROlg+orurwLxzUCQJrwzZvl48fK3xog/bxMqnuYonPayvbZL3QaAAA7hXEEQL7zqV8ofvlJaf06Wav2skZnyKIodBYAYBe5uzT5U8XD+kiL5ieeIXVZZ1nteqHTAADYJYwjAPKN526QD+op/+BtqWo1RTc9INtrn9BZAIBd5O7SjCmKh/SR5syUquytqOMdUv2jZWah8wAA2GWMIwDyhc+crrjnE9KyxbJTW8haXCLLzg6dBQDYRT5zuuKhfaXvvpYqVOZYXgBAWmIcAbBLPG+jfPir8tFDpQqVE1eLHFg3dBYAYBf53FmKh/VLPDuqTDnZRe1lJ5zG8A0ASEuMIwB2ms+ZmbhaZMFc2YlNZRdcJiueEzoLALALfP5cxW+8Kk0eL+WUkp17qazxmbJixUOnAQBQYBhHAOwwz9sof3OA/J0hUplyijrfI6vbIHQWAGAX+KL58jf7y7/4SCpWXNasleyUs2Q5pUKnAQBQ4BhHAOwQn/WN4t5PSovmyY47WXbB5bKSvHEGgFTlSxbK33pNPuF9KTtb1vRc2annyEqVCZ0GAEChYRwBsF08d4N8WD/52Dek8pUUdblPdlD90FkAgJ3kSxfJR7wm/3ScVCRL1qS57PTzZGXKhU4DAKDQMY4A+Es+bZLifs9Iy5fITjpDdl5bni0CACnKly6SjxosHz9WskjW6ExZ0/Nk5SqETgMAIBjGEQBb5atWyAe8mLj/vMreim55SFbzoNBZAICdkLhSZKB8wjjJLPEg7dPPl5WvGDoNAIDgGEcA/IHHsfzjd+Wv95Y25srOvlh22nkc3wgAKciXLJCPGJQYRaIisoanJ64UYRQBAGALxhEAv+NzZynu96z0w7dSzbqKLukk26Nq6CwAwA7yhT/JRw6Sf/ahlJX1/7fPnCsrxygCAMD/YhwBIEnydWsSD1x9f5RUqrTs8htkR58kMwudBgDYAT5vtvytgfLJ46XsoonjeE9pwTNFAADYBsYRIMN5HCv+ZGziFpo1q2UnnS5r0VqWw/G8AJBKfPZ3ikcMlKZ+LhUvkbh15pSzZaXLhk4DACDpMY4AGcxnfaOfH+4lnzlDql5TUed7ZfvWCJ0FANhO7i59O03xyEHSjKlSTilZ84sSx/KWZOQGAGB7MY4AGchXLpe//kri4XzlKyVuoTmqoSyKQqcBALaDx7E0baLiUYOlWd9IZcvLzm8na3gaR60DALATGEeADOK5G+Sjh8nfGSJt3iQ74wJVbHO1fl67LnQaAGA7+KZN8okfyd8eIs2fI1XcTXZxB9nxJ8uyi4bOAwAgZTGOABnA483y8e/Jh/WTfvlZOuxYRedfJqu8h6ISORLjCAAkNc/NlX/yrnz0MGn5EmnPfWRX3CA7/ARZFm/nAADYVXw3BdKYu0vTpyge3Eua96O034GKOtxR+SexAAAgAElEQVQm27926DQAwHbw1avk40bIx70lrVkt1ail6KKrpYMbcCskAAD5iHEESFM+6xvFQ/tI306TKu0ua3+r7PDjOJoXAFKAL10kH/OG/ON3pY25Ur0jFTU9V7Z/ndBpAACkJcYRIM34/LmKh/WRpnwmlS4ru6i97ITTZNnZodMAAH/BZ38vHz1UPmm8FEWJh2Wfdo5sz31CpwEAkNYYR4A04Yvmyd96Tf75h1LxErIWbRJHORYvEToNALANHsfSV18ofneY9N1/pBIlE4NI42ay8hVD5wEAkBEYR4AU54sXJEaRzz6QsrNlp54ja3qurFSZ0GkAgG3w3A3yT9+Tv/uGtGSBVKGy7MIrZMefwnG8AAAUMsYRIEX5wnnyUYPln70vZWXJTjk78ZPGMuVCpwEAtsFXLpePGyn/4G1p7Wqp2gGy9rfI6h/DyTMAAATCd2AgxfjcWYpHDpImf5q4UqRx88SVImXLh04DAGyD//h94iGrEz+W4liqf7SiU86WatTmYdkAAATGOAKkAHeXvp+ueNRg6etJUokc2ennJ54pwpUiAJC0fPNmacoExWPekGbOSDwTqtGZiX/tViV0HgAA+H+MI0AS83iz9OUExe8MlWZ/J5Uqk3jQaqMzZDmlQucBALbCV6+Sfzxa/v5I6edlUuU9ZBdeKTvuZFkJnicCAECySYpxpEePHpo8ebLKli2rRx99VJLUp08fTZo0SVlZWdp9993VqVMnlSxZMnApUDh8w3r5+LHyMW9ISxdJu1WRte4gO6aJrFix0HkAgK3wuT/I33srcXJY3kapdj1FF7WXDjlCFhUJnQcAALYiKcaRk046SU2bNlX37t23/Nohhxyiiy++WEWKFFHfvn01dOhQtWnTJmAlUPB8yUL5uBHyT8ZI69dJNWopOr+ddOiRvKkGgCTlm/Lkkz+VjxuRuHWmaDHZsY1ljZrJ9tondB4AANgOSTGO1KlTR0uWLPndr9WrV2/LP9esWVMTJkwo7Kzfib/4WBv3219eYXcemoZ85XEszZiq+L23pGkTpSiSNThe9n/s3Xd8VfX9x/H392SQhLDC3rKRPaUgICKKA8VfVbQOXHVvsVY7HNVatVZtq2jrHhVXRaoiIssJOAAVEFEEFQEhQCAQyDqf3x9HYhACGTf33Jv7ej4eeZjmnNzzzrc5OR8+95zvd+Qxch26hh0PAFAGy9mobW++LH/aS9LWnODRmXHnyg05TK42jz4CABBPYqI5sj+zZs3SkCFDytw+Y8YMzZgxQ5J0++23q1GjRhE9vvm+sl9+SpvXr1XyAR2VPvp4pQ0fLS+Dx3yqS3JycsT/f4w1xTmbtHPmq9rx5v9U/MMaefUaKP2ks5U+eqySshpHLUcijHUsYbyjh7GOnlgd6+quD3Z8+oG2vvCYUvsNVsbRJyq1z0FynhfRY2B3sfq7VhMx1tHDWEcPYx098TjWzsws7BCStH79et1xxx0lc47s8tJLL2nFihW65ppryn3Hxpo1ayKez3bmqfaSBcqd+qL07ddSrTS5g4bLHTxKat+Fu0kirFGjRsrOzg47RsSZXyx9/qns3TdlC+dJxUVSl55yw0fL9R0sl5IS9Uw1daxjFeMdPYx19FTnWLdo0SJirxXp+sAKC9TAmXKSmQsqWjivo4exjh7GOnoY6+iJl9qgtJi+c2TOnDn6+OOPdcMNN4TefHBpGcoYfby29ztYWvWV7O1psvlvyd6ZLjVrFdxCO3iEXP2GoeZEbLIf1sjenyWbO0vanC3VrhM8NjN8tFyzVmHHAwBUgktJVXKjRhKFNgAAcS9mmyOLFi3SlClTdPPNN6tWDK3O4ZyT2nWSa9dJdvK5sg/fDVYVeekJ2eSnpO595QYdItdnkFxaethxESLL3Sr7+D3Z/Lekr5ZKzpO695U37hyp96BQ7hIBAAAAAOwpJpoj9957r5YuXarc3FxdeOGFGjdunCZPnqyioiLdcsstkqROnTrp/PPPDznp7lxahtywI6RhR8jWfR80SebPkT1ytyy1VtAgOegQqXsfuWT+IZwILH+nbNH8YAnHJQuk4mKpeWu5X46X+8Whcg24swgAAAAAYk1MNEeuvPLKPb42cuTIEJJUnmvWUu6X42XHny6tWBY0ST56L/hHcnptuT4HyfU/WOrWRy4lNey4iCDbkSf79EPZgvelxR9LBQVSViO5UWPlBh0itTog9MfCAAAAAABli4nmSE3iPE/q1E2uUzfZKedJSxfJPn5ftmiebO5sKS1drtdAqc8gue795FjxJi7Z1s2yTz6UffKBtGShVFQo1cuSO/hwuQEHSx27sWIBAAAAAMQJmiPVyCWnSL0GyvUaKCu6WFr2mWzB+7JF86UP3pYlJUmde8j1GijXa4Bck+qZdRdVZ2bS96uChsinH0orl0tmUlZjuRFHyfUfIrXvSkMEAAAAAOIQzZEocckpUo9+cj36yU6/SPp6uezTD2SLPpA997DsuYelxs3kuveV695X6tJLLj0j7NgJzXK3yJYukpYsDP67ZVOwoV1nueNOletzkNSSR2YAAAAAIN7RHAmB85KkjgfKdTxQ+uWZsvVrZUsWyJYslM2dLZvzupSUJB3QSa5zD7muPaUO3eRiaNWemsjytknLl8i++Ez2xWfS6lXB3SG168h16xOsRNS9n1z9rLCjAgAAAAAiiOZIDHBNmss1OUY69BhZUWEwoeuShcE/0t94Sfb6i1JSsnRAR7mOB8p1OFDq0EWuboOwo8ctM5M2bZB99bn09RfBf7/7OmiGJKdIHbrKHfcrue79pLYdgoYWAAAAAKBGojkSY1xyitSlp1yXnpIk25knffW5bNlnsq+Wyma+IntjcrBz42Zy7ToH/3hv20lq055Hccpg27ZK36yQrfpS9s1XwZwhOT8+JlMrLbhLZ8wpwV067TqzohAAAAAAJBCaIzHOpWVIPfrL9egvSbLCQunbFbKvPpet+Fz21dJgcldJck5q2kJq2Vau5QFyrQ6QWraVGjVNmIlCrbhY+uF72fffSKu/kX2/Kng8ZuP6n3Zq0lyuc4/g0aYOXYN5Q5K4MwQAAAAAEhXNkTjjUn585KNDV0n/J0myrTnBXRHffCn75mvp269lC+YGj45IUkqq1KS51KylXNNWUtMWco2bSY2aSvUaxF3jxIqLpc3Z0sb1svVrg2bIuu+lH9ZIG9ZJxUXBjp4nNW0p176LdOjRcm06BHfZZGSG+wMAAAAAAGIKzZEawNWtL/XsL9ezf8nXLH+ntOZb2epV0rrVQfPgu1WyhfMk35ft2jE5WcpqLDVsIlcvS6offLj6WVJmPSmzjlS7jpRZJ3jkpxpZYYG0LVfavlUFa1bJX/1t8OjLlk1SziZZzqbgDpDN2ZLv//SNyclSkxZS81ZyfQdJzdvItWwrNW8dNJMAAAAAANgHmiM1lKuVFsyd0a7zbl+3okIp+wcp+wdZ9vqg2ZD9g2zTBtmXS4JmRHHRT82T0mqlS+npUlp68HlaupRaS0pJkUtKDpoUySnB4z27HdSkwsLgdYsKpaIiKX+ntHNH8JG/Q9qRJxXkl3zL5tLfn5IaNG3qZQUr/DRsKjVqIteoaXD3S8PGTJgKAAAAAKg0miMJxiWnSM1aSc1aye1lu5kFd29s2SjlbpX9eCeHtm0Nvv5jU8N2NTa25khFhbKiImlX42OP1ooLGicpKcGqO8kpUq1awSM9TVsETZa09B/vUKkrl1lX9Vq20hZfQVMkvbbczxsuAAAAAABECM0R7MY5J9WpG3xIe22gRENqo0Zy2dkhHR0AAAAAkEjiayZOAAAAAACACKM5AgAAAAAAEhrNEQAAAAAAkNBojgAAAAAAgIRGcwQAAAAAACQ0miMAAAAAACCh0RwBAAAAAAAJjeYIAAAAAABIaDRHAAAAAABAQqM5AgAAAAAAEhrNEQAAAAAAkNBojgAAAAAAgIRGcwQAAAAAACQ0miMAAAAAACCh0RwBAAAAAAAJjeYIAAAAAABIaDRHAAAAAABAQqM5AgAAAAAAEhrNEQAAAAAAkNCcmVnYIQAAAAAAAMLCnSMVcN1114UdIWEw1tHDWEcX4x09jHX0JPJYJ/LPHgbGO3oY6+hhrKOHsY6eeBxrmiMAAAAAACCh0RwBAAAAAAAJLemmm266KewQ8aR9+/ZhR0gYjHX0MNbRxXhHD2MdPYk81on8s4eB8Y4exjp6GOvoYayjJ97GmglZAQAAAABAQuOxGgAAAAAAkNBojgAAAAAAgIRGcwQAAAAAACQ0miMAAAAAACCh0RwBAAAAAAAJjeYIAAAAAABIaDRHAAAAAABAQqM5AgAAAAAAEhrNEQAAAAAAkNBojqBK/vCHP8g5pzvvvHOv23//+9/LOacjjjiiytuOPPJIOec0a9asyP0AAAAgYqpSF1QGtQEAIFJojqDSVq9erbvvvluNGzfWJZdcstd9Fi5cKEnq27dvlbfddNNNkqRrrrlGvu9XOvfGjRv12GOP6fTTT1e3bt1Uu3Zt1apVS61atdLxxx+vyZMnl+t12rRpI+fcHh8ZGRnq1q2bJkyYoA0bNlQ6Z3UpK3dmZqZ69eql3/3ud9q0aVPYMcsUL+N+1lln7TVneT4GDRoUdvwSOTk5+sc//qGjjjpKLVu2VFpamurUqaMOHTpo+PDh+v3vf69Zs2ZV6ZyMpOo+v+PlPAHCUNW6oDKoDSIj3v/mxdO4J/J1itoIMc+ASho/frxJsr/+9a9l7jNixAjr0qWLTZkyJSLbRo8ebZLsiSeeqHTu5ORkk1TykZaWZrVr197ta0cddZRt3769zNfIzs4u2bdBgwbWtGlTa9q0qTVs2HC312nZsqV9++23lc4aaWXlbtCgwW6527RpY6tXrw477h7iadxHjBhRkq/0R2ZmZknOvW1v2rSpXXvttaFm3+X555/fY2zr1q27x/lSr149830/7LhmVr3nd7ycJ0BYqloXVBa1QdXE+9+8eBv3RL5OURsh1tEcQaWsXr3akpKSLDU11TZu3Bi1406ePNkkWffu3Sv9GpLsoIMOsokTJ9qKFStKvr5y5Uo799xzS/6onX766WW+xhtvvFGy36JFi3bbtmnTJrvwwgtLtp9xxhmVzhpppXN/+umnu23Lzs62Sy65pGT7mWeeWeXj3XjjjSbJbrzxxiq/lln8jntp5513nkmy1q1bR/y1Iznejz/+uDnnTJIdeuihNnXqVNuyZUvJ9jVr1tikSZNs7NixdsQRR1T5eJES6fM7GucJUBOEVReYURtUFbVBdHGd2lN11UaR/l2L19oI5UdzBJWy64/N8ccfH9Xj5ufnW1ZWlkmyd999t1KvMWvWrH1uv+CCC0ouKGW9w3DbbbeZJEtJSbH8/Pw9thcXF1v37t1NkjVs2LBSOfenuLjYZs2aZatWrSr39+zKnZ6ebkVFRXts933fevfuXdK5r6pIX5TiddxL69+/v0mysWPHRjhZ5MZ73bp1Je/i/PrXv97v/nl5eVU63r5UdLwjeX5H6zwBaoKw6gIzaoOfH4PaYHexNu418ToVq7VRJH/XYqk2QvVhzhFUmJnpkUcekSSdeuqpZe43YcIEOed07LHHRmxbamqqTjjhBEnSv//970rlP/TQQ/e5/dxzzy35/KOPPtrrPgsWLJAkdevWTampqXts9zxPAwcOlBQ8W1pcXFyprHuzdOlSXX/99TrggAM0cuRIrVy5stzfuyt37969lZSUtMd255x69+4tSdq8eXNkAkdQvI77LkVFRVq8eLEkqV+/fhHLFmkvvPCCtm3bJkm6+eab97t/enp6xDNUdrwjeX7H63kCRFtV6oJvv/22ZE6B2bNn7/M4V111lZxz6tevn8ys5OvUBtQGUvyMe026TlEbla06aiNUv+SwAyD+LF68WKtXr5YkDRs2rMz9Fi1aJGnvk65VdpskDR8+XA899JCmTZtWseDllJaWVvJ5WRfQjz/+WJJKLjx7U1BQIEnKzMzc64WrItavX69JkybpqaeeKjm2JDVo0EBZWVnlfp1d3ztgwIAy9/nuu+8kSa1bt65k2uoTr+O+y5IlS5Sfny8pcpMRVodly5aVfF6Zn7OyIj3ee1OR8ztezxMg2qpSF7Rp00b169dXTk6OlixZUuY/HFeuXKmJEydKku666y4553bbTm1AbRBv416WWL9OURtFtzZCdNEcQYW9/fbbkoI/ts2aNStzv+pqjuyarXr9+vVatmyZunbtWoH0+zdnzpySz3v27LnH9s2bN5d0xsu6EJuZ5s6dK6nyXfAdO3ZoypQpeuqppzR9+nQVFRVJCi6aY8aM0Wmnnaajjz56r++S7E3p3P3799/rPu+++67eeustSdKJJ55YqdzVJV7HvbRd7/RUJV801KtXr+TzW2+9VTfeeKNSUlKq5VjVOd57U5HzOx7PEyAMVa0LevbsqXfeeUdLliwp83uvv/56FRQU6JhjjtHIkSP32E5tQG0QT+O+L7F4naI2im5thBCF+EgP4tQZZ5xhkmzMmDFl7rNq1aqS5yVXrlwZkW2l7Xrm79FHH63CT7KnzZs3W/PmzU2SDRs2bK/7zJgxoyTjzJkz97rPAw88ULLPI488Uu7j+75vs2fPtnPOOcfq1q1b8hqe59nIkSPtkUce2W3ip4oonfvnk3etW7fObrvttpJx7dSpU6WPU1okn/WM13Ev7bLLLjNJ1qRJkyq/1t5Earw//PDD3WbTb9y4sZ111ln25JNPRmSm/2iN989V9PyO1nkCxLuq1gW7Jo8s67ycP3++OecsKSnJlixZUuYxqA0qjtqgbGFcq2LpOlUTaqNI/q5Vd22E2EBzBBV22GGHmSQ7++yzy9xn18zx9evXj9i20tq2bWuS7NZbb634D1CG4uJiGzNmjEmyWrVq7THj+S533nlnyR/G7Ozskq/n5+fb4sWL7eqrrzbP80ySjRw5cq8TZf3c0qVL7frrr7c2bdqUvLYk69Onj9111132/fffV/nnK527SZMmJUuj1alTZ7djHn744RFb9i2SF6V4HffSDj74YJNko0ePjujr7hLJ8X7hhResdevWu43Lro+ePXva3XffXeHJxqI93qVV5vyO1nkCxLuq1gUPPvigSbKsrKy9fu/w4cNNkl1wwQX7zEFtUHHUBnsK61oVK9epmlQbRXry3+qojRBbaI6gwnr16mWSbMKECWXuc8MNN5gULHMVqW2l9e3b1yTZFVdcUfEfoAyXXnppud5ZOPnkk/f6R/HnH6eeeqrl5ubu97jPPvvsbt93wAEH2O9+97t9vjtWGfvL7Xme3XHHHRV+3WeffbbMNel3rfleu3btMvd57733IpI/Vsd9l+Li4pJ3cq6//vpKv060xtvMrLCw0KZNm2aXXXaZ9ejRo2T5ul0f3bt3t61bt5Y7dzTH++cidX5X9jwBarKq1gXvv/9+yTm2du3a3bZNmTLFJFlmZqatW7dunzmoDSqO2mDP3GFdq2LhOhWPtVE06yKzyNZGiD00R1BhnTt3Nkn2u9/9rsx9jj32WJNkV199dcS2lTZkyBCT9v8uUnlNmDCh5I/aPffcs899O3XqZJIsIyOj5A9rs2bNrGPHjjZs2DC75pprbOHCheU+9mOPPVZy7PT0dLv22mv3uE0yEnblvvLKK0u+lp+fb0uXLrVTTz215II6Y8aMCr1u6fyV+Zg9e3aF8sfbuO/y+eeflxzvhRdeqPTrRGu892b9+vX20EMPWYcOHUpe75prrqlw7miMd2mVOb8jfZ4ANVlV64Lc3NySf2CUPreKioqsa9euJsluueWW/eagNqg4aoOyc0fzWhUr16l4rI3CrIvMqlYbIfbQHEGFDR482CTZRRddVOY+rVq1Mkn25JNPRmxbabuKpX0VYuX1m9/8puSP2V//+td97rtly5aSAu7OO++s8rHNgudDb7755pKL3a6PHj162F/+8pdKrxlfWuncjz/++B7bfd8vWWP+kEMOqfLxdonU7YzxOu6lPfPMMyXHWLFiRURfe5dI3z5alm+//dZSUlJMkg0aNKhc3xPt8d6lsud3NM8TIN5VtS4wM2vXrp1Jsr///e8lX9s1V0TLli1t+/bt+81BbVAx1AZ7CuNaFUvXqZpWG0WrLjKrXG2E2ENzBBV23HHHmSQ7+eST97o9Ozu75A/dZ599FpFtP9ekSZNyddf355prrik5ZnkurHPmzCnZf9asWVU69t7MnTvXLr74YmvYsGHJcZxzNnToUHvggQd2e562IkrnLusZ1ieffLJkn6+//roqP0aJSF2U4nXcS9v1u1avXj3zfT8CqfcUzSKgffv2JQVTRUVjvM2qdn5H8zwB4l1V6oJdxo4da5Ls/PPPN7PgbpKmTZuaJHvsscfKlYPaoGKoDfYtmrVBLF6nakJtFM26yKxqtRFiA80RVNh1111nkmzgwIF73f7mm2+aJEtLS7PCwsKIbCtt69atJZNrTZs2rdI/R+lbGMv7jsPdd99dcnHIycmp9LH3p6CgwCZPnmy//OUvrVatWiU5U1JSbMyYMTZp0qRyvYv289ypqalWUFCw1322bNlS0vGO1Dswkbooxeu4lzZy5EiTZCNGjIhw6p9EqwjYvn27ZWRkmCQbN25cpV+nOse7Kud3tM8TIN5VpS7Y5Y9//KNJsoMPPtjMfpqjpHfv3lZcXLzfDNQG1AbVpbquVfFynYrn2iiazZFI1UYIF80RVNjUqVNL/jDv3Llzj+27ZtHeW5FU2W2lzZo1yyRZcnJyuSbX2pvSF6S77rqr3N932mmnmSTr2LFjpY5bGZs2bbIHH3ywZDbvXR+ZmZl2+umnl2vW8F25+/Tps8/9Dj30UJNkgwcPjkj2SF2U4nXcS8vKyjJp//PpVEVVx/uTTz4p18918cUXl4zH9OnTK3Wsn4vkeFf1/I72eQLEu6rUBbu88MILJgWr2axZs6ZkIsU333yzXBmoDagNoiFS16p4vU7FW20Uid+1MGsjRJ8noIIOPvhgJScnq6CgQIsWLdpj+8KFCyVJffv2jdi20ubPny9J6tevnzIzMysWXtJvf/tb/e1vf5Mk3X333ZowYUK5v3fBggWSpAEDBlT4uJXVoEEDXXDBBXr33Xe1YsUK3XTTTerYsaO2bdump59+WsuXL9/va+zK3adPn33ud8wxx0iS5s2bp3Xr1lU9fITE67jvsmrVKm3atEnS/n+/w/Twww+rQ4cOOvPMMzV16lTl5OSUbNu0aZOmTJmioUOHauLEiZKkSy+9VIcffnhEjh2p8Y7E+R2v5wkQlqrUBbv07NlTkpSTk6PzzjtP27dv11FHHaVRo0aVKwO1AbVBNERi3OP5OkVtFN3aCCEIuzuD+LTr2eC9TXq2a0K0Bx54IGLbSts18du9995b4dzffPNNSVfX87wyl/Xa9VF6Yqxt27aV3LK7vwmzouH999+3iy66yD788MN97lc69/7GrPSs4Q8++GCVM0aiYx+v417af//735Jxrc7lAKs63sOGDdvtnSBJVqdOnZJ3cHd9pKen21/+8pfIhi9DRcY7Uud3tM8ToCaobF2wS3FxsaWnp5ecW0lJSbZ48eJyH5/aIEBtEI7yjntNvU7Fam0Uid+1WKyNUH2SI9dmQSK54IILNGXKFD3zzDO69dZb5ZyTJOXl5ZV0jX/e1a7sttJWrlypuXPnKj09XePHj69wbt/3d/v8hx9+2Of+27ZtK/l80aJFJd/fv3//Ch870gYPHqzBgwfvd7/Suff3TkPXrl3Vvn17ff3113r55Zd1wQUXRCRrVcTruJe2613TjIwMdenSpTpiRcTUqVM1depUzZw5UwsWLNCqVau0ZcsWeZ6nZs2aqXv37ho1apTGjx+vFi1aRCVTRcY7Uud3PJ4nQNgqUxeU5nmeunfvro8++kiSdM4556h79+7lOja1wU+oDcJR3nGvqdcpaqPo1kaoPs7MLOwQiD++76tz585asWKF3nrrLQ0fPjwqx/3Tn/6kG2+8UWeffbYeffTRqBwTAADsW1h1gURtAACIDJojqLRJkybp1FNP1VFHHaWpU6dW+/G2b9+uAw44QLm5ufriiy/Utm3baj8mAAAon2jXBRK1AQAgcpiQFZV2yimn6KCDDtLrr79eMhFadbrvvvuUnZ2tyy+/nOIHAIAYE+26QKI2AABETtJNN910U9ghEJ+ccxo4cKCaNWumBg0aqHPnztV6vEWLFmnQoEG69tprlZaWVq3HAgAAFRPtukCiNgAARA6P1QAAAAAAgITGYzUAAAAAACCh1cilfNesWVMtr9uoUSNlZ2dXy2tjd4x19DDW0cV4Rw9jHT3VOdaRXBqxOuoDfs+ii/GOHsY6ehjr6GGsoydeaoPSuHMEAAAAAAAkNJojAAAAAAAgodEcAQAAAAAACY3mCAAAAAAASGg0RwAAAAAAQEKjOQIAAAAAABIazREAAAAAAJDQaI4AAAAAAICERnMEAAAAAAAkNJojAAAAAAAgodEcAQAAAAAACY3mCAAAAAAASGg0RwAAAAAAQEKjOQIAAAAAABIazREAAAAAAJDQaI4AAAAAAICERnMEAAAAAAAktOSwAwAAAAAAEA1mJuXvkHK3SjvypJ07pPwdsp07gs+LiqSiwp/+6/t7vkiSJyWnSMnJP/43RS4tXSr9kZ4pZdaRUmvJORf9HxQVRnMEAAAAABDXzEzK2yZl/yBtXC/bvEnaslHK2STL2SRtzdGGvG3yt26Riosif/yyNiSnBE2SzLpSvQZy9bOkeg2l+llyDRpKjZpIDZvKpWdEPBMqhuYIAAAAACAu2Lat0rrVsh/WSOu+l637XtqwVtq4Prjzo7SkZKleA6l+ltS4uWo1aqydyalBoyKzjlx67Z/u9KiVLqWlSSnBnSBKSpFSkuW8pD0zFBf/dGdJcaFUUCDl7wyO/+OH5W2TtuVK27dK27F+1pMAACAASURBVHKD3Fs2y77/Vtq6WfL93RsqGZlSoyZyjZtLzVpKzVrKNW0pNW0pl1G7WscUAZojAAAAAICYYkVF0ppvZN+tkr5fJfv+G2n1Kmlrzk87JSVLTZpLTZrLde0lNWwi16ip1LCx1KCxVDtTzvtpms26jRqpIDu7ytlcUpKUlCTVqlX2Pvv62fzi4LGeTdnSpvWy7B+k7OC/9t3X0sK5uzdPshpJLQ+Qa9U2+G/rdkHzZC+NG1QezREAAAAAQGjM94O7Qb7+QvrmK9k3K6TvVgZ3ZkhSSqrUoo1cj/5SyzZyzVtLTVsEj6MkxV+DwHlJwR0t9RpI7Trt0UixokJpwzrph+9la1dLq7+Rfb9KtnSRVFwUNE1qpUmt28sd0FFq21GufRepcTPmN6kCmiMAAAAAgKixgnxpxTLZjx/6epmUtz3YmJ4htekgN3KMdEDH4C6JJs0T6i4Jl5wiNW8tNW8t1+enr1tRYfAo0XcrgybSqi9lb0+TCgqChkmdelKHA+U6dpXrcGAwfskpYf0YcYfmCAAAAACg2lhhgfT1F7Jln8m++FRauTyYs8O54I6QAUOlDl2Dux+atNjtURj8xCWnSK0OkGt1gDT4UEk/zn+y9jvZ18ukrz4PGk6L5gXNktRaUsducl16yHXpKR3QKS7vtIkWmiMAAAAAgIgxs+AxmSULZEsWSssXB5OWOk9q015u5LFyXXpIHQ+Uy8gMO25cc0lJPzVMhh8pSbKtOdKXS2VffCZbvlg2+amgWZJeWzqwl1z3vnLd+8k1bBJm9JhDcwQAAAAAUCVWWCB98Znskw9kn34kbdoQbGjaUm7oEXLd+kidutEMiQJXt77Uf4hc/yGSfmyWLF8sW7ooaFgtmBs0S5q1lOs1UK73QcHjOAl+VwnNEQAAAABAhdn2bbJP5ss++UBasjBYzja1ltStj9zRJwV3KDRqGnbMhOfq1pcGDJUbMHT3u3o+WyCb+aps+stS7TpyPfrJ9f2F1KO/XK20sGNHHc0RAAAAAEC5WO7WYE6LBe9Ln38iFRdL9RvK/WKEXO9BUteecimpYcdEGZxzwUSvzVtLo8bKduZJSxYFTa7PPpLNf0tKTZV6DJDrP0Su1wC5tIywY0cFzREAAAAAQJlsZ55swTzZB28FDRHfD5aNHTVWrv/BwaooLCEbl1xaRskjOOYXB3OVfPx+8OjNgvdlySlSr4HyBg2Xeg6o0Y0vmiMAAAAAgN1YUZG0+GPZ/LeCx2YKC6SGTeRG/zJYXaZ1OxoiNYzzkqQuPeW69JSdcp709TLZR+/JPnxH/oL3pfTacv0Gyw06JNivhq0qRHMEAAAAACBJstWrZO/NlM2fI+VukTLryg0dJTdohNS+Cw2RBOE8L1gGuGM32UnnSF98Kpv/tuzj92TvzQgaZYNHyg0ZKde4WdhxIyImmiMTJ07UggULVK9ePf3tb38r+frrr7+uadOmKSkpSf369dPpp58eYkoAAAAAqHksb3twh8h7M6RvvpKSkqXeB8kbcpjUva9cckz8sxEhcUlJUre+ct36yk67ULZoftBAe+052avPSp17yA09PJijJLVW2HErLSZ+y0eMGKEjjzxS999/f8nXFi9erI8++kh33XWXUlJStGXLlhATAgAAAEDNYqu+lL01TfbB21JBvtSqndwp58kddIhcnbphx0MMcqm15A4aLh00XLZpg2zubNl7M2SP3iN79iG5IYfJDR8tNWoUdtQKi4nmSLdu3bR+/frdvjZ9+nSNHTtWKSkpkqR69eqFEQ0AAJTB1q6WzX5N2xo2kkafEHYcAEA5WEG+7MN3ZLNek75dIf34j103/EgmVkWFuKzGcseMkx11ovTFZ7K335DNfk02Y4o2de8rO+lcuZZtwo5ZbjHRHNmbtWvXatmyZXr22WeVkpKiM844Qx07dtzrvjNmzNCMGTMkSbfffrsaVVOXKjk5udpeG7tjrKOHsY4uxjt6GOvqYcVFyv/wPe14/b8q+PQjKTlFGn18TI51NOoDfs+ii/GOHsY6eqI11sXZPyhv2mTtmD5FlrtFSa3bKeO8CUo7ZLS82pnVfvxYwO91NWpymDTsMBXnbNLOma9q5+zXldWmrZIaNAw7Wbk5M7OwQ0jS+vXrdccdd5TMOTJhwgR1795dZ599tlasWKF77rlH9913X7k6mWvWrKmWjI0aNVJ2dna1vDZ2x1hHD2MdXYx39DDWkWW5W2XvTpfNmSptypayGskNP1Ju2BFq3L5jtY11ixYtIvZa1VEf8HsWXYx39DDW0VPdY20rl8vemCxbOFcyBXOJHDYmWG0kwe4S4fc6eho2bKiNGzdWy2tHsjYoLWbvHMnKytKgQYPknFPHjh3leZ5yc3NVty7PvgEAEC327deyWa8Gz6MXFkhdeso7+Typ90HBBG0AgJhjvi99+qH86ZOlL5cGS7COGit36NFyjZqGHQ8JIB4bbzHbHBk4cKAWL16s7t27a82aNSoqKlKdOnXCjgUAQI1nfrG06AP5M/8nLV8ipaYGy/WNPEauZduw4wEAymBFhcEEmdMnS+u+l7Iay518brCSSFpG2PGAmBYTzZF7771XS5cuVW5uri688EKNGzdOI0eO1MSJEzVhwgQlJyfrkksuicvuEwAA8cLytgczzs98Rdq4XmrYRO7Es4KiujZvUABArLL8/ODRxzcmS5uzpTYd5M67Rq7/wdzlB5RTTDRHrrzyyr1+/fLLL49yEgAAEo9tWCeb+Yrs3RlS/g6pUzd5J50j9RlEUQ0AMcx25slmT5W9OUXK3RL8/R5/qdS9L28sAxUUE80RAAAQXWYmrfhc/ptTpIXzJc+TGzhUbtRxcm33vjocACA22M4dwZKp0ydL23KlHv3kHXWSXOfuYUcD4hbNEQAAEogVF0sL58qf/rK0crmUkSl31AnBJH3142e5PQBIRJa/M2iKvDFZ2rZV6jlA3rG/kmvXKexoQNyjOQIAQAKw/J2yd2fIZkyRsn+QmjSXO+3CYKLVWmlhxwMA7IMVFcrefkP26nPB4zM9+gVNkfZdwo4G1Bg0RwAAqMFs62bZzNdkc6ZKedukDl3ljTtX6j1QzmM+EQCIZeYXy+a/LZvyn2Ci7M495F38O7mOB4YdDahxaI4AAFAD2fo1sjdelr0/UyoukvoMknfE/1FQA0AcMDPps4/kv/Sk9P03UpsO8k6/mIlWgWpEcwQAgBrEVn0pf9p/pQVzpaRkuSEj5Q4/Xq5Zy7CjAQDKwb5dIf+Fx6RlnwaPQJ5/rVz/IXKeF3Y0oEajOQIAQJwzM2nZp/Jff1H6/BMpvbbckSfIHXasXL0GYccDAJSDbcqWvfy0bN5sqXam3Cnnyx0yWi45JexoQEKgOQIAQJwy35cWzQ+aIqu+lOo1kDvxLLnhR8qlZ4QdDwBQDpa/U/7/Jsne+K/km9zoXwariGVkhh0NSCg0RwAAiDNWXCz74G3Z6y9Ka7+TGjeTO+PiYOWZlNSw4wEAysHMZB+9p+zJT8g2/CA3YKjcCWfKNWoadjQgIdEcAQAgTlhhgey9mbI3XgqW423ZVu7XE4KCOomVZwAgXth3K+U/+29p+RIlHdBJOusKuc49wo4FJDSaIwAAxDjLz5e9M032xmQpZ5PUrrO8k38t9RrIBH0AEEcsb7vsf8/IZr0WzCty+sXKOv5X2rh5c9jRgIRHcwQAgBhlO/Nkc16XTX9Zyt0ide4h7+wrpQN7s5QjAMQRM5PNf0v24mPS1hy5Q46UO/4MudqZ3PkHxAiaIwAAxBjbkSeb9arszSnS9lypW195x4yT69w97GgAgAqytavl/+cB6YvPpLYd5V3yB7l2ncKOBeBnaI4AABAjLG+7bNYrsjf/J+Vtk3oOkDfmZLn2XcKOBgCoICsslL3+ouz1F6TUWnKnXSQ3/Ag5jztFgFhEcwQAgJBZ3nbZzFdkM6ZIedul3gfJO/YUubYdw44GAKgEW75Y/lMTpXWr5Q4aLnfyuXJ1G4QdC8A+0BwBACAktiNPNvN/weMzedulPr+Qd+zJcm06hB0NAFAJlrdd9uJjsnemSw2byLviRrke/cOOBaAcaI4AABBltjNPNvPVYKLVvG3BnSLH/YqmCADEMfv0w+BukS2b5Y74P7njfiVXKy3sWADKieYIAABRYvk7ZbNfk73xkrQtV+o1MGiK8PgMAMQt27ZV9tzDsnlzpJZt5V38OyZcBeIQzREAAKqZFRbI3pome/1FaWuO1KOfvONOo3gGgDhnC+fJf+p+KW+b3LGnyB19klxyStixAFQCzREAAKqJFRXJ3pshe/U5KWej1LWXvIuuk+vYLexoAIAqsLxtskkPyebNltq0l3fVn+Ratws7FoAqoDkCAECEmV8s++Bt2f8mSRvWSR26yjv3KrmuvcKOBgCoIluyUP7j/5C2bv7xbpFxcsn8swqId5zFAABEiJlJC+fJn/Ifac23Uut28i6/QerRX865sOMBAKrA8nfKXnhU9tY0qXlreZf+njmjgBqE5ggAABFgn38if/JT0srlUrOW8i64Vuo3RM7zwo4GAKgiW/Wl/IfvltavCVaiOf40uZTUsGMBiCCaIwAAVIGt+jJoiixdJDVoJHfmZXKDR8olJYUdDQBQReYXy17/r+yVSVLdBvIm3CrXpWfYsQBUA5ojAABUgv2wRjb5KdnH70mZdeTGnSs34ijeSQSAGsKyf5D/yD3SV0vlBg6TO+0iudqZYccCUE1ojgAAUAG2ZbPs1Wdl70yXklPkxpwc3GKdnhF2NABAhNhH78p/8n5JJnfuVXKDRjB3FFDD0RwBAKAcbGee7I3JsukvS8VFcsNGB42Reg3CjgYAiBDLz5c991DQAG/XWd5518g1bhZ2LABRQHMEAIB9sKIi2TtvyF55VsrdIjdgqNz/nS7XpEXY0QAAEWTffyP/X3dKa7+TO/IEubGnsUQvkEA42wEA2AszkxbMlf/Sk9L6NVLnHvIuu0GuXaewowEAIsjMZO++KZv0byk9Q95VN8t16xt2LABRRnMEAICfsRXL5L/wqLRimdSijbzL/ij1HMDz5gBQw1j+TtnTD8jmzZYO7C3v11fL1eVxSSAR0RwBAOBHtn6t/JeekD5+X6qXJTf+Urkhh7EsLwDUQLbmW/kP3iGtWy039lS5o0+S8/h7DyQqmiMAgIRn23Nlrz4nmz1VSkqSO/ZXckccL5eWHnY0AEA18OfOlj09UaqVJu+qP8kd2DvsSABCRnMEAJCwrKhQNuf1YLLVHXlyBx8WTMBXPyvsaACAamCFhbLnH5bNeT2YS+q8a/ibD0ASzREAQAIyM+mTD+S/8Fgw2eqBveWNO0euVbuwowEAqoltypb/4O3SyuVyo38p939n8NgkgBI0RwAACcVWr5T/3CPSsk+lZq3kXX6D1KM/k60CQA1mn38i/6G7pIICeRdeJ9d/SNiRAMQYmiMAgIRgW3NkU/4je+dNqXZtuVMvkBs2Wi6ZSyEA1FRmJpv+suy/T0jNWsq76Hq55q3CjgUgBlERAgBqNCsslM16Rfba81JBvtxhY+TGnCJXOzPsaACAamT5+bIn75N98JbUb4i8sy+XS8sIOxaAGEVzBABQI5mZ7JMP5D//iLR+rdRroLyTzpZrxjuGAFDT2cYN8if+WfpupdzxpwfL9PL4JIB9oDkCAKhxbO13ypn4Z/kL5wfzilxxk1yPfmHHAgBEgS1fLP/BO6SiQnmX/kGu18CwIwGIAzRHAAA1huVtk73yrGzWqypMy5A7+Vy5EccwrwgAJAj/7WmyZ/4lNW4m75Lfc7cggHKjWgQAxD3zfdl7M2STn5K2bZUbdoQanXO5NhUWhx0NABAFVlwse/4R2axXpR795Z13jVxG7bBjAYgjNEcAAHHNvv5C/qR/S6u+lDp0DR6hadtBXr0GUnZ22PEAANXM8rbJ/9dfpaUL5UaNlTvpLDkvKexYAOIMzREAQFyyrTmyl56QvTdTqpcld+5VcoNGMOEeACQQ+2GN/PtukTaskxt/qbxhR4QdCUCcojkCAIgrVlwsm/O6bMp/pIKdcqP/T27MySzPCAAJxpYvkT/xNslJ3lW3yHXpEXYkAHGM5ggAIG7Y8iXyJ/1LWr1KOrC3vF9dINecyfYAINH482bLnvin1KipvMtukGvSPOxIAOIczREAQMyzrZtlLzwumzdbymok78LrpH6DeYQGABKMmQWrkr0ySerSU95F18nVrhN2LAA1AM0RAEDMMr9Y9tY02eSnpYJ8uaNOlDtmnFyttLCjAQCizAoLZU/eJ5s3W27wSLnxl8glp4QdC0ANQXMEABCTbOVy+U8/IH27gkdoACDBWd42+RP/In3xmdzY04JGOXcPAoggmiMAgJhi27fJXnpS9s4bUr0Gcuf/Rm7AUIpgAEhQtmmD/L/fLP2wRu7cq+T94tCwIwGogWiOAABigpnJ5s6WvfiYtC1X7rBj5Y47VS6dVWgAIFHZ6pVBYyR/p7wrbpQ7sHfYkQDUUDRHAAChs7Xfyf/Pg9IXn0ntu8i78ma5Nu3DjgUACJF9/kmwVG9ahrxr/yLXql3YkQDUYDRHAAChsYJ82dQXZNNekmqlyZ1xidzQw+U8L+xoAIAQ+R+8LXv0Xqlpi+COkazGYUcCUMPRHAEAhMKWLgwmXN2wTu4Xh8qddLZc3fphxwIAhMyf+Yrs2YekTt3kXfoHuYzMsCMBSAA0RwAAUWVbN8uee1T2wVtS05byrr6FZ8gBAMHcUy8/LZv6gtTnF/LOmyCXWivsWAASBM0RAEBUmJns3TdlLz4uFeyUO/YUuaNOlEtJDTsaACBkVlwse3qi7N035YYdIXfaRXJJSWHHApBAaI4AAKqdrV0t/+n7peVLpM7d5Z1+iVzzVmHHAgDEACsskP/QXdLCeXJjTg5WKmP5dgBRRnMEAFBtrKhQ9vp/ZVOfl1JryY2/VO7gUUy4CgCQJNnOPPkT/yJ9/oncKefLO2xM2JEAJCiaIwCAamFffS7/yfuktd/JDRwmd8qv5eo2CDsWACBG2Lat8v/xJ+mbr+TOuUre4EPDjgQggdEcAQBElO3Ik01+UjbndalBI3mX3yDXc0DYsQAAMcRyNsq/50Zp/Vp5F10n1+cXYUcCkOBojgAAIsY++VD+fx6QcjbKjRwjd/zpcmnpYccCAMQQ27BO/t1/lHK3yrviRrmuvcKOBAA0RwAAVWe5W2TPPiT74G2pZVt5F/5Wrn2XsGMBAGKMrVst/29/lAry5U24Ra5d57AjAYAkmiMAgCowM9n8t2TPPSTt2BGsMHDUCXLJKWFHAwDEGFu9KrhjRJL3mz/LtWoXciIA+AnNEQBApdimbPlPT5Q++0hq30Xe+MvkWrYJOxYAIAbZqi/l33uTlJIq7+pbWM4dQMyhOQIAqBAzk70zXfbiY1JxsdzJ5wbzi3hJYUcDAMQg++pz+f+4WcrIlDfhVrnGzcKOBAB7oDkCACg327BO/lP3S59/InXpKe/MyyhyAQBlsuWLg+V66zcM7hjJahR2JADYK5ojAID9Mt+XzZ4qe+kJyfPkzrhYbugRcp4XdjQAQIyyZZ/K/+ctUsMmQWOkflbYkQCgTDRHAAD7ZOvXyH/in9LyJVKPfvLOuEQuq3HYsQAAMcyWLpJ//61S4+byrv6TXN0GYUcCgH2iOQIA2CvzfdmsV2WTn5SSUuTOukJuyEg558KOBgCIYbb4Y/n33yY1axncMVKnXtiRAGC/aI4AAPZg69fIf+wf0ldLpZ4DgrtFGjQMOxYAIMbZZx/Ln/hnqXnroDGSWTfsSABQLjRHAAAlfppb5HEpOUXu7CvkBnO3CABg/2zxx/In3ia1aBM0RmrXCTsSAJQbzREAgKQfV6J5/B/S8sVSj/7yxl/K3SIAgHKxJQuDR2mat6IxAiAu0RwBgARnZrK3pslefExyTu7My+QOHsXdIgCAcrGlC+Xf/2epGY0RAPGL5ggAJDDblB2sRLN0oXRgb3lnXi7XkJVoAADlY59/Iv++P0tNWzDHCIC4RnMEABKQmcnmzZFN+rdUXCR32oVyhxzF3SIAgHKz5Yvl33eL1KS5vKtvlatDYwRA/KI5AgAJxrbmyH9qorRontTxQHlnXyHXpEXYsQAAccRWLJP/j1ukrCY/LtdLYwRAfKM5AgAJxBbNk//k/dKO7XInni13+HFyXlLYsQAAccS++Ur+32+W6tWXN+EWubr1w44EAFVGcwQAEoDlbZc997Ds/ZlS63byJtwq17Jt2LEAAHHGVq+Uf8+NUkbt4FpSn1XNANQMNEcAoIazLz6T/+i90uaNckePkzv2ZLnklLBjAQDijK1dLf/uG6TUWkFjJIsJvAHUHDRHAKCGssIC2ctPy96cIjVuJu+3t8t16Bp2LABAHLIN6+Tf/QfJuWCOkcbNwo4EABFFcwQAaiBbvVL+w3dL338jd8iRciedI1crLexYAIA4ZDkb5d9zg1RYKO83t8k1axl2JACIOJojAFCDmO/L3pwie/kpKSNT3mV/lOs1MOxYAIA4Zblbg0dptm5hvioANRrNEQCoIWzThmBukS8+k/r8Qt74S+Tq1As7FgAgTlnedvl/v0nK/kHeFTfKtesUdiQAqDY0RwCgBvA/eFv2nwek4mK58ZfKDT1czrmwYwEA4pTl75R/3y3S6pXyLvm9XJeeYUcCgGpFcwQA4pjlbZdN+pds3hypXWd5v75arkmLsGMBAOKYFRUp56+/l776XO6838j1HBB2JACodjRHACBO2VdLg0lXN2XLHXtKsExvMn/WAQCVZ74ve/zvKvh4rtzpF8sbODTsSAAQFTFRRU+cOFELFixQvXr19Le//U2S9Pzzz2vmzJmqW7euJOlXv/qV+vXrF2ZMAIgJVlwse/U52WvPSw0bs0QvACAizEz23MOy+W8p87QLtOOQI8OOBABRExPNkREjRujII4/U/fffv9vXjznmGB133HEhpQKA2GMb1sl/5G5pxTK5Xxwqd+oFcukZYccCANQA9tpzslmvyo0aq4wTxmvHxo1hRwKAqImJ5ki3bt20fv36sGMAQEzz580JJl11ntyvJ8gbdEjYkQAANYQ/e6psyjNyg0fKnXQ2k3oDSDgx0RwpyxtvvKG3335b7du31/jx45WZmRl2JACIOtuRJ3vmwWDS1Y4Hyvv1BLmGTcKOBQCoIezj92ST/iX1PkjuzMvkPC/sSAAQdc7MLOwQkrR+/XrdcccdJXOO5OTklMw38txzz2nz5s26+OKL9/q9M2bM0IwZMyRJt99+uwoKCqolY3JysoqKiqrltbE7xjp6GOvoquh4Fy5fqi1336DiDetUe9zZqn3imXJJMd3Xjhn8bkdPdY51ampqpb83GvUBv2fRxXhHXsHiBdp881VK6dhVDW76u1ytNEmMdTQx1tHDWEdPrNYG+xKzFXb9+vVLPj/ssMN0xx13lLnvqFGjNGrUqJL/nZ2dXS2ZGjVqVG2vjd0x1tHDWEdXecfbfF/2xmTZlKelelnyrrlNOzt1087NOVFIWTPwux091TnWLVpUfmnqaNQH/J5FF+MdWfbdSvl/vV5q3EzFF16njbnbpNxtkhjraGKso4exjp5YrQ32JWabI5s3b1aDBg0kSR988IFat24dciIAiA7L2ST/0Xukzz+R63+w3BmXyNXmsUIAQORY9g/y/36TlJYh78qb5GrXCTsSAIQqJpoj9957r5YuXarc3FxdeOGFGjdunJYsWaJVq1bJOafGjRvr/PPPDzsmAFQ7W/yx/EfvlfJ3yI2/VG7o4UyKBwCIKMvdKv/em6TCAnnX3iGX1TjsSAAQuphojlx55ZV7fG3kyJEhJAGAcFhRoWzyU7LpL0st28o7/89yLdqEHQsAUMNYfr78+26RNm2Qd9Wf5FpyrQEAKUaaIwCQyGz9Wvn//qv0zVdyI46SO+kcudRaYccCANQw5hfLf/hv0srl8i78rVynbmFHAoCYQXMEAELkf/iu7Kn7JOfkXXSdXL8hYUcCANRAZiZ79mFp0Ty5U87jegMAP0NzBABCYAX5sucekb09TWrfRd5518g1ahp2LABADWXTX5bNfk3uiOPlHXZs2HEAIObQHAGAKLO138n/153S99/IHXmC3NjT5JL5cwwAqB7+h+/IXnxMbsBQuRPOCjsOAMQkqnEAiKIds1+X/+CdUq00eVfcKNejf9iRAAA1mC1fInv0HqlTN7lzrpTzvLAjAUBMojkCAFFg+fmySQ9q63szpc495J03Qa5+w7BjAQBqMFv3vfyJt0kNm8q75PdyKalhRwKAmEVzBACqma35NniMZu13qn3S2dpx2HFySUlhxwIA1GCWu1X+P/8UTPh9+Q1yteuEHQkAYhrNEQCoRv77s2T/eSB4jObKm5U5fJR2ZmeHHQsAUINZYYH8iX+WNmXLm3CrXJPmYUcCgJhHcwQAqoEV5Msm/Vv27ps/PkZzjVz9rLBjAQBqOPN92WN/l776XN4F18p1PDDsSAAQF2iOAECE2brV8h+8I1iN5uhxcsf9isdoAABRYVOekX34jtwvz5QbMDTsOAAQN2iOAEAE+R++I3viPiklmdVoAABR5c+dLZv6vNzQw+WO/GXYcQAgrtAcAYAIsMJC2QuPyma/JnXoKu/838hlNQ47FgAgQdiXS2VP/lPq0lPutAvlnAs7EgDEFZojAFBFtnF9sBrNyuVyh48NbmVO5s8rACA6bMO6YMnerCbyLrpOLjkl7EgAEHeo3gGgCuyzj+U/crfkFwcFab8hYUcCACQQy9su/5+3SL4v77I/smQvAFQSzREAqATzi2WvPCt79TmpVTt5F/1WrkmLsGMBABKIFRcHdy6uXyPvypvlmrUMOxIAxC2aIwBQQZa7Vf7Dd0lLF8kdfJjcqRfKpdYKOxYAIMHYC49KSxfKjb9UrmuvsOMAQFyjOQIAgOro6QAAIABJREFUFWArlgXv0uVukRt/qbxhR4QdCQCQgPy335DNfEVu1FiuRQAQATRHAKAczEw2Z6rsuUekBg3lXXenXNsOYccCACQgW75Y9syDUve+cieeFXYcAKgRaI4AwH5Yfr7s6ftl8+ZIPQfIO/dqudqZYccCACQg27BO/gO3S42bBcvGJyWFHQkAagSaIwCwD7Z+TVCEfv+N3NjT5I4+Sc7zwo4FAEhAtjNP/v1/DlZIu/SPchk06gEgUmiOAEAZ7JMP5D9yj+R58i6/Ua5Hv7AjAQASlPl+cE1a+528K26Ua8oKaQAQSTRHAOBndlumt00HeRddJ9eoadixAAAJzF55Vlo0X+6U8+S69Q07DgDUODRHAKAU254r/+G7pcUfB8v0nnaRXEpq2LEAAAnMFrwve/VZuSGHyY0cE3YcAKiRaI4AwI/su5XyH/iLtClb7vSL5YaPlnMu7FgAgARmq1fJf/ReqV1nudMv4roEANWE5ggASPLnvyV78p9SRqa839wm16Fr2JEAAAnOtufKn3iblJYh7+LruZMRAKoRzREACc2Ki2UvPi6bMUXq3F3eBdfK1W0QdiwAQIKz4mL5/7pT2pwt75rb5P6fvTsPsLHs/zj+uW4zhhn7vpUQlZ9S0aaSSmhVlBbtT4ulorSqSCptSgtRqbRZ6qFEq0rrU1kSEmUpZGRvhjFmub+/P6a0icGcc53l/fqLcZs+vibnms+57uuuVNV3JABIaJQjAJKWZf9atPBcMEfuuFPkzrpULoV/FgEA/tn4UdJ338hddDW7GQEgCvguAEBSsp8WFW1Vztogd0kvBa2O9x0JAABJUvjVx7J3X5M79iQFR53gOw4AJAXKEQBJJ/ziQ9nzQ6XyFRTcfJ9c/b19RwIAQJJky5fIRj0m7b2fXJf/+I4DAEmDcgRA0vjr+SLNfjtfpJLvWAAASJJs00aFwwZJZTMUdLtZLiXVdyQASBqUIwCSgm3MUvjkA0X3b3O+CAAgxlhYqPDpwdK6NUVPTavI4eAAEE18ZwAg4dnyJQofv1v6dZ3cxdcoOLKt70gAAPyFTRwtzZ0h17U7B7ACgAeUIwASms34TOEzQ6T0DAU3DJJruI/vSAAA/IV985Vs8ji5I9vKHdPBdxwASEqUIwASkoWh7I3RskljpYb7KOh+i1ylKr5jAQDwF7ZqhcKRD0t7NpLr2k3OOd+RACApUY4ASDiWm1O00Jz1ZdG7cF27y6VyqB0AILbYli0Kn7hXck5Bt5vkUkv7jgQASYtyBEBCsVWZCofeLa1cLnfO5UWHr/IuHAAgxpiZ7KVh0s8/Kbi6n1z1Wr4jAUBSoxwBkDDsu28UDr9PkhT0HiC3X3PPiQAA2Db76G3Z/z6UO/Ucuf1b+I4DAEmPcgRA3DMz2dQ3ZWOekmrWVXDVbXI1avuOBQDANtmS72Vjn5KaHSx3yjm+4wAARDkCIM5ZQb5s9FOyj9+WDjhEwWV95Mqm+44FAMA22casol2OFaso+M91ckHgOxIAQJQjAOKYZWcpHH6v9P1cuQ6d5c44Xy4o5TsWAADbZGFYdGB41noFN90nV66C70gAgN9QjgCIS/bzUoWPD5Q2rJP7z7UKDj/WdyQAALbL3npVmjtD7rxucns19h0HAPAnlCMA4o7NnqbwqQeltDIKbrhHruE+viMBALBdNn+27PWX5Q45Wq7Nib7jAAD+hnIEQNwwM9l7r8lefU7ao6GCnrfKVanmOxYAANtlG9YVlfo168hd2JNHzANADKIcARAXrCBf9uITss+mSC1aKbikt1xaGd+xAADYLissLCpGcjcruO4uuTIcGg4AsYhyBEDMs+wshU/cI/0wT+6Us+VOPZfT/QEAccEmji46OPzSa+Xq7uk7DgDgX1COAIhplrlM4WMDpfVr5S7ro+CwY3xHAgCgWGzuTNlbr8gddYKCIzg4HABiGeUIgJhl336tcMT9Umqqguvvlmu0r+9IAAAUi21Yq/CZh6U6e8qdc4XvOACAHaAcARCTwg/flI15Uqqzp4KrbperWt13JAAAiqXonJHB0pZcBVfeKJeW5jsSAGAHKEcAxBQrLJSNGyn7YJJ0wCEKLu/D4XUAgLhib/x2zsglveVq7+E7DgCgGChHAMQMy81R+OSD0pzpcm07yp11sVxQyncsAACKzeZ9LXvzFbkjj1fQ6jjfcQAAxUQ5AiAm2NrVCh+7U8pcJnd+DwXHdPAdCQCAnWK/rlf49ENSrXpy53bzHQcAsBMoRwB4Z0t+UPj4QCk/X0Gv/nJND/IdCQCAnWJhqHDkQ9KWzQr63M05IwAQZyhHAHhlMz5X+MxDUoXKCvrcJVdnT9+RAADYafbOeOm7b+QuvEquLq9lABBvKEcAeGFmsncnyP47SmrQREHPW+UqVPIdCwCAnWaL5stee1HukKPljjrBdxwAwC6gHAEQdVZQIBs9QvbxO3Itj5K7pJdcabYfAwDij23aqPCpB6Uq1eXO7yHnnO9IAIBdQDkCIKosZ5PCEfdL876WO+ksuY5d5YLAdywAAHaamSl8/jFpw1oFN90nl57hOxIAYBdRjgCImq1PpFm5XO6iqxWw9RgAEMfso7elmf+TO/MSuQZNfMcBAOwGyhEAUWE/LVL42EApL1fBNf3lmh7oOxIAALvMfl4qGzdSanqQ3AkdfccBAOwmyhEAEWezpyl88gEpo1zRtuO69X1HAgBgl1l+nsKnHpDKlFVwaW9uDwWABEA5AiCiwqlvyl5+UtqjgYKrb5erVMV3JAAAdou9+pz0808KruknV7Gy7zgAgBJAOQIgIiwMZeNHyd6ZIB1wiILLr5crU9Z3LAAAdovNnib7YJLc8afK7d/SdxwAQAmhHAFQ4iw/T/bMENn0T+XanCh3zhVypUr5jgUAwG6xDesUPvuIVK+BXOeLfccBAJQgyhEAJco2Zikceo+0cJ7cmRfLtTtDzjnfsQAA2C0WhgqfHVJ0sPgV18ulpvqOBAAoQZQjAEqMrV6p8NEB0ppf5K64QcEhR/uOBABAibD335DmzZI7v4dc7T18xwEAlDDKEQAlwn78QeGjd0qFhQquvVOuSTPfkQAAKBG2fIls/CjpwMPkWrf3HQcAEAGUIwB2m82ZrnD4fVL5igp69ecdNQBAwrC8LQqfGixllFdw4dXcKgoACYpyBMBuCT9+R/bSE1K9vRRc3Y9H9QIAEoqNf15asVRBrzvkylfwHQcAECERLUe6d++utWvXSpIOOugg3XLLLf967fjx4zVmzBiVK1dOzzzzTCRjASgBZiab+LJs0lip2cEKrrxRrky671gAAJQYmztD9v4bRY/tbXaw7zgAgAiKWDmSnZ29tRiRpDlz5ignJ0fp6dv+5mnJkiWSpAYNGkQqEoASYgUFsheHyj57X+7I4+XO7ymXwkY0AEDisOxfix7bW7e+XOeLfMcBAERYEKlPvHjx4q0/LleunAoKCjR9+vQdXk85AsQ2y92scOjdRcXIKefIXXQNxQgAIKGYmcLnH5dyNiq47Dq51NK+IwEAIizi5Uj58uV17LHHSpK+/PLLbV67ceNGrV69WpLUsGHDSEUCsJssa4PCB2+Vvv1a7oIeCjqex8F0AICEY59NkWZ9KXfGBXL1eOMOAJJBxMqR32+TadiwoQ477DBJ0qxZs5Sbm/uPa/+8y4RyBIhNtmqFwntvlDKXKujZV0HrDr4jAQBQ4mz1StmYp6V99pdr29F3HABAlER850iDBg3UuHFjValSRfn5+Zo5c+Y/rv29SClbtqxq1qwZqUgAdpH9+IPCe2+SNm9S0OduueaH+o4EAECJs7BQ4TMPS4FTcElvuSBiS2UAQIyJyL/4Gzdu1KpVqyQVlSPOOR16aNE3U1988cU/rv+9SGnYsCFb9IEYY3NnFt1KUzpNwU33yzXcx3ckAAAiwt6ZIC38Tu68K+WqVvcdBwAQRREpR37fCSL9ccDq77fWfP3118rLy9vm9RzGCsSW8IsPFT4+UKpeW8HN98vVqus7EgAAEWFLF8lef1muxZFyh7XxHQcAEGURLUcyMjK23iaz3377qUKFCtqyZYtmzZq19dqcnBz98ssvkjhvBIgl4TsTZCMflvZuquCGe+QqVfEdCQCAiLD8PIVPPySVqyB3QQ92MgNAEopIOfL7bTJ77bXX1heXIAjUsmVLSX+9tWbJkiUyM0nsHAFigYWhwleekb36rFzLoxT0ukMuPcN3LAAAIsYmvCBlLlNw8TVyGeV9xwEAeBDRnSN/LzsOP/xwSdKMGTNUUFAg6Y8ipUyZMqpdu3Yk4gAoJisokD07RPbua3LHnix3+fVyqam+YwEAEDG2YK5sykS5NifKNTvYdxwAgCclXo7k5ORo5cqVkv55m0yzZs2Unp6uzZs3a/bs2ZL+KFL22msvBZwIDnhjW3IVDr1L9sVUudPPlzv3Ck7pBwAkNMvNUfjsEKlaTbkzL/EdBwDgUYl/5/Pjjz/+620yKSkpatGihaQ/bq358yN/Afhh2VkKB98mfTtL7sKrFJzchfutAQAJz8Y9I61breDS3nJpZXzHAQB4VOLlyI5uk/n9qTXTp0/Xxo0blZmZKYnDWAFfClevVHj/TdLyHxX0uFnB0e18RwIAIOJsznTZJ+/KtTtDbu+mvuMAADxLKelP+OfDWLd1m8yBBx6otLQ0bdy4UZMnT+YwVsAjy1ymdY8MkHI2Kuh9h1yTZr4jAQAQcbYxS+Gox6S69eU6dvUdBwAQA0p858i/Hcb6u9KlS+uggw6SJE2ePHnrx+rVq1fSUQBshy1eoPC+m6XCAgU3DKIYAQAkDXt5hLQxW8Gl13LwOABAUgmXI7m5uVqxYoWk7e8E+f3WmtzcXElS/fr1OYwViCKbO7PojJH0DFUeNEJuD3ZuAQCSg03/VDbtE7lTzpbbk9u6AQBFSrSR+PNhrNs7Q+Tggw9W6p9aem6pAaInnPaJwsfvkmrUUXDTfUqpVdd3JAAAosKyNih8abhUf2+5E8/0HQcAEENK9MyRfffdV+PGjdvhdWXLltVLL71Ukv9pAMUQTn2zaCvx3vspuOp2ufQM35EAAIgKMysqRnJzFFzSW65UKd+RAAAxJGbuZRk2bJguu+wy9enT5x+/NnHiRHXp0kVZWVkekgHxz8wUThore2m4tH9LBb0HUIwAAJKKTftEmvm53Gld5eru6TsOACDGxEw50qZNG/Xt2/cfH1+zZo3mzJmjatWqeUgFxD8LQ9m4kbLXX5I7vI2C7rfIlU7zHQsAgKixX9cX7Zxs0ESu3em+4wAAYlDMlCNNmzZVuXLl/vHxUaNGqWvXrnLOeUgFxDcrLJQ996hsykS540+Vu6S3XEqJP8EbAICYZWYKXxwmbcnldhoAwL+K6e+Spk+fripVqmivvfbyHQWIO5afp/DJB6RZX8p1PE/u5LMpGQEASce+nFr0WnjWJXK16/mOAwCIUTFbjmzZskXjx4/XbbfdtsNrp0yZoilTpkiS7r333ojdgpOSksLtPVHCrHdPuHmTfh00QHlzZqj85dcp/aR/P5GfWUcX844eZh09sTrraKwPYvXPnqh2dt6F69Zo7dinlbpPM1U++1J2jewEvrajh1lHD7OOnnictbPfn70bA1atWqX77rtPgwcP1tKlS3XnnXcqLa3obIS1a9eqcuXKGjRokCpVqrTdz7NixYqI5KtWrZrWrFkTkc+Nv2LWu842Zil8ZIC0dJHcxb0UHHHsdq9n1tHFvKOHWUdPJGddp06dEvtckVgf8HUWXTszbzNTOGyQNHeGgn6PsGtkJ/G1HT3MOnqYdfTEy9rgz2J258iee+6pp59+euvPe/bsqUGDBqlChQoeUwGxzTasVfhQP2n1yqKDVw88zHckAAC8sGmfSLO+kDvzYooRAMAOxUw5MmTIEM2bN0/Z2dnq1q2bunTpouOOO853LCBu2OqVCh+6XcrOUtCrv9y+B/iOBACAF5a1QTb6t6fTnNDRdxwAQByImXKkd+/e2/31oUOHRikJEH9sxVKFD/eT8vMV9Bko16CJ70gAAHgTvjxcyt2s4OJr5ALOGQEA7FjMPMoXwK6xnxYqfOAWyUzBDfdQjAAAkprN+Eya8bncKefI1dnTdxwAQJyImZ0jAHaefT9X4WMDpYzyCq4bKFejtu9IAAB4YxuzFL40XKq/t1yHzr7jAADiCDtHgDhlc2cofOQOqVJVBTfeSzECAEh6NvZpKWejgouv5rG9AICdws4RIA7ZzM8VPvmgVGcPBdfeKVe+ou9IAAB4ZXOmy76YWnQ7Tb0GvuMAAOIM5QgQZ8L/fSh77hGpQRMF1/STSy/nOxIAAF7Z5hyFLwyTau8hd9JZvuMAAOIQt9UAcSSc+qbsmYelJs0U9B5AMQIAgCQbP0rasLbo6TSpqb7jAADiEDtHgDgRvjNB9uqzUvNDFVx5o1xqad+RAADwzhbMlU19S65tR7mG+/iOAwCIU5QjQIwzM9kbY2RvjJZreZTcf66TS+F/XQAALG+Lwucfk6rXkju9q+84AIA4xm01QAwzM9l/nysqRlodL3d5H4oRAAB+YxNHS6syFVzQUy6tjO84AIA4xndZQIyyMJSNeVL24ZtybU6SO/cKuYA+EwAASbKfFsnee03uqBPk9mvuOw4AIM5RjgAxyMJC2fNDZZ9NkWt/hlzni+Wc8x0LAICYYIWFRbfTlK8od+YlvuMAABIAb0MDMcYKCmQjHy4qRk49h2IEAIC/sSmvS0sXKzj3SrkMntwGANh97BwBYogV5Ct88gHp6y/kOl+koENn35EAAIgptipT9vrL0oGHSQcf4TsOACBBUI4AMcLy8xQ+ca80Z7rcOZcrOP5U35EAAIgpZqbwxWFSSoqC87qxsxIAUGIoR4AYYFtyFQ69W5o/W+6CHgpad/AdCQCAmJP74ZvSd9/Ide0uV7mq7zgAgARCOQJ4Zrk5Ch8bKP3wndzFvRS0Os53JAAAYo5lrVf2s49KezeVa93edxwAQIKhHAE8spxNCh8dIC35Xu7yPgoOOdp3JAAAYpKNHSnLzVVw4VU82h4AUOJ4ZQE8sU0bFT7cT/pxoYIrb6IYAQDgX9jcGbKvPlZG5wvkatfzHQcAkIAoRwAPLDtL4eBbpeVLFHS/RY7T9gEA2CbbskXhS8OlWnWV0flC33EAAAmKcgSIMsvaUFSMrPxZQc/b5Jof4jsSAAAxy94YLa35RcEFPeVSS/uOAwBIUJw5AkSR/bpe4eDbpLW/KLj6drn9mvuOBABAzLKli2XvvSZ31AlyTZr5jgMASGCUI0CU2Ia1RcXI+rUKrrlDbh8WeQAA/BsLCxW+MFTKKC935sW+4wAAEhy31QBRYOvWKHzgVmn9OgW9KEYAANgR+/At6ccf5M6+TC6jvO84AIAEx84RIMJs7eqiM0ayf1Vw7QC5Rvv6jgQAQEyz9Wtlr70gNT1I7tDWvuMAAJIA5QgQQbbmF4UP3irlbFJw3UC5Bk18RwIAIOaFY56SCgsVdO0m55zvOACAJMBtNUCE2OqVRcXI5k0KrruTYgQAgGKw2dOkmZ/LndxFrkZt33EAAEmCnSNABGwtRnI3K7juLrn6jXxHAgAg5tmWLQpfHiHV3kOu/Rm+4wAAkgg7R4ASZqsyFT7YV9qSq6DPQIoRAACKySaPkdauUnB+d7mUVN9xAABJhJ0jQAmyVSuKnkpTkKegz11yezTwHQkAgLhgP/8ke/c1uVbHyzXhqW4AgOiiHAFKyB/FSH5RMVKPYgQAgOKwMFT44jCpbLrcmZf4jgMASELcVgOUgL8WIwMpRgAA2An22RRp4XdyZ14iV76C7zgAgCREOQLsJooRAAB2nWVnyf47Stq7qdwRx/mOAwBIUpQjwG6gGAEAYPfYhOeLHnt/fne5gKUpAMAPXoGAXUQxAgDA7rGF38k+eVeubUe5uvV9xwEAJDEOZAV2QdHjem/746k0FCMAAOwUKyxU+NITUuVqcqee4zsOACDJsXME2Em2eqXCwbdK+VsUXEcxAgDArrAPJ0nLf1RwzmVyZcr6jgMASHKUI8BOsNUrFT54q7Rli4JrB8rtQTECAMDOsvVrZa+9LDVrIR10hO84AABQjgDFZWt+UTj4Nil3s4Lr7pTbs6HvSAAAxCV75RkpLFRw3pVyzvmOAwAA5QhQHLZ2ddGOkc2bfitGGvmOBABAXLJ5s2TTPpE78Uy56rV8xwEAQBLlCLBDtm5N0RkjOZsU9L5Trv7eviMBABCXrCBf4egRUvVach06+Y4DAMBWlCPAdtiGtUW30mT/qqD3HXINGvuOBABA3LL3Jkorf1Zw7pVyqaV9xwEAYCvKEeBf2K/ri4qRX9cr6HWHXMN9fEcCACBu2brVskljpAMPl9u/he84AAD8BeUIsA2WtaGoGFm/VkGv/nJ77+c7EgAAcS0cN1KSKTj7P76jAADwD5QjwN/YxiyFD90urf1FwdX95Bo39R0JAIC4Zt9+Lc34XO7Es+Sq1fQdBwCAf6AcAf7ENm1U+HA/aVWmgqtul9unme9IAADENcvPVzj6SalGbbn2Z/iOAwDANlGOAL+xnE0Kh/SXVixV0OMWuf2a+44EAEDcs/dek375WcG5V3AIKwAgZlGOAJIsN0fhowOkZUsUdLtFrhkHxQEAsLts3WrZ5HFFh7Dy2goAiGGUI0h6tiVX4aN3Sku+V3DFDXLND/EdCQCAhGCvPicZh7ACAGIf5QiSmuVtUTj0bmnhfLnL+sgdfITvSAAAJASbP1s27RO5Dp05hBUAEPMoR5C0LD9f4fD7pPmz5S7ppeCQo31HAgAgIVhBgcIxT0lVa8h16OQ7DgAAO0Q5gqRkBQUKn3xAmjNd7vweCo441nckAAAShk19U/r5JwVnXyZXOs13HAAAdohyBEnHwkLZMw9Ls76QO/cKBa3b+44EAEDCsKz1sokvS00Pkg48zHccAACKhXIEScXCUPbcY0X3QJ95iYLjTvEdCQCAhGLjn5fy8hSce7mcc77jAABQLJQjSBpmJnt5uOx/H8h1PE9B+zN8RwIAIKHY4gWyz96Xa3uaXK16vuMAAFBslCNICmYmG/eM7KO35U7sLHfy2b4jAQCQUCwMFY5+UqpYWe6ULr7jAACwUyhHkBTs9ZdkU16XO/5UuTMuZJsvAAAlzP73ofTjD3KdL5Yrk+47DgAAO4VyBAkvnDxONnmc3NHt5M6+jGIEAIASZptzZONHSQ33kTvsGN9xAADYaZQjSGjhlImy116UO+wYufO7U4wAABABNmmslLVBwTlXyAUsLwEA8YdXLySs8JN3ZWOflg46XO6S3nJBKd+RAABIOLbyZ9n7b8gd2VauQWPfcQAA2CWUI0hI4ZcfyV4YKjU7WMHlN8iVohgBACASwnEjpdKl5Tpd4DsKAAC7jHIECcdmfSF75mGpcVMF3W6RS031HQkAgIRks6dJc6bLnXK2XIXKvuMAALDLKEeQUGze1wpH3C/V31vB1bfLpaX5jgQAQEKygnyFY0dKterKHXeK7zgAAOwWyhEkDPthnsKhd0u16ino1Z/HCAIAEEH2wSRp1QoFXS6TS2GXJgAgvlGOICHYT4sUPnanVKmagmsHyGWU9x0JAICEZVkbip5Q06yF3P4tfMcBAGC3UY4g7lnmMoVD+ktlMxRcN5B7ngEAiDB7/SUpb4uCLv/xHQUAgBJBOYK4ZqtXKnzodikIioqRqtV9RwIAIKHZsiWyT96Ta3OSXO16vuMAAFAiKEcQt2zDWoUP95Py8hRce6dczTq+IwEAkNDMTOHYp6WMDLlTz/UdBwCAEkM5grhkG7MUPtRPyvpVQe875Ort5TsSAACJb+b/pAVz5Dp2lcso5zsNAAAlhnIEccc25ygccoe0eqWCq26Va9DEdyQAABKe5ecpfOUZqW59uaPb+44DAECJohxBXLG8LQofv0tatlhBt5vk9j3AdyQAAJKCTZkorV2l4OzL5EqV8h0HAIASRTmCuGEFBQpH3C/98K3cpdfKNT/UdyQAAJKC/bpeNvkVqfmhcvs19x0HAIASRzmCuGBhKHv2EWn2NLnzrlRw2DG+IwEAkDTs9ZekgjwFZ17iOwoAABFBOYKYZ2ay0U/KvvpI7owLFLQ5yXckAACShi1bIvv0PbljT5arVdd3HAAAIoJyBDHPXn9JNvVNufadFJx0lu84AAAkDTNTOG6klF5O7pRzfMcBACBiKEcQ08J3X5NNHid3dDu5zhf5jgMAQHL55itp/my5087l0b0AgIRGOYKYFX42RfbKM3ItjpQ7v7ucc74jAQCQNKwgX+Erz0q16sm17uA7DgAAEUU5gpiU++VHslGPS00PkvvPdXIBjwwEACCabOqb0qoVCrpcKpeS4jsOAAARxSsdYo7Nn61fHxkgNWisoMctcqmpviMBAJBUbGOW7I0xUtODpGYtfMcBACDiKEcQU+zHHxQ+frdK1a4nu6afXFoZ35EAAEg6NmmstHlz0a4RbmsFACQByhHEDFu5XOEjA6Ry5VW5/xCtNxZjAABEm/2yougpcUe1latb33ccAACigjNHEBNs3WqFD/eTnFNw7Z0qVbW670gAACSl8L/PSSml5Tp29R0FAICooRyBd5adpfDh/tLmHAW975CrWcd3JAAAkpJ9P1f6+gu5EzvLVazsOw4AAFFDOQKvLHezwsfulNb8oqDnbXJ7NvIdCQCApGRhqHDcM1LlanJtO/qOAwBAVFGOwBsryFf4xCDpp4UKrrxRbp9mviMBAJC07KuPpJ8Wyp1xgVxamu84AABEFeUIvLAwlD0zRJo3S+7Cq+QOPMx3JAAAkpblbZFNeEGqv7fcYcf4jgMAQNRRjiDqzEw25knZtE/kzrxYwZFtfUcCACCp2ZSJ0ro1Cs66VC5geQgASD4x8yjfYcOGaebMmapYsaIGDx4sSRozZoymT58u55zIwbq/AAAdnUlEQVQqVqyoHj16qEqVKp6TYnfZpLGyD9+Ua3eGgvadfMcBACCpWdYG2VuvSs0P5RZXAEDSipm3Btq0aaO+ffv+5WOnnXaaHnzwQT3wwAM6+OCD9eqrr3pKh5ISTn1LNvFluSOOkzvzYt9xAABIejZpjJS3RQGvywCAJBYz5UjTpk1Vrly5v3wsPT1964+3bNki51y0Y6EE2YzPZS8Plw44RO6iq/n7BADAM1v5s+zjd+Rat5erVc93HAAAvImZ22r+zejRo/Xxxx8rPT1d/fv39x0Hu8jmz1b49INSw30UXHGjXKlSviMBAJD0wvGjpJTScqee4zsKAABeOTMz3yF+t2rVKt13331bzxz5swkTJig/P19dunT5x69NmTJFU6ZMkSTde++9ysvLi0i+lJQUFRQURORzJ7L8xQu0/raeCqrVVJW7n1BQvsIOfw+zjh5mHV3MO3qYdfREctalS5fe5d8bjfVBPH+d5c37Rutv7a6M8y5XubMu8R2nWOJ53vGGWUcPs44eZh09sbo22J6Y3znyu6OOOkr33nvvNsuRtm3bqm3bP554smbNmohkqFatWsQ+d6KyVZkK77tJKpsuu+p2rduSJ23Z8QyZdfQw6+hi3tHDrKMnkrOuU6fOLv/eaKwP4vXrzMwUjhwiVaqiza3aKTdO/gzxOu94xKyjh1lHD7OOnlhdG2xPzJw5si2ZmZlbfzx9+vSIDQGRYVnrFQ7pLxUWKug9QK5KNd+RAACAJM34TFq8QK5jV7m0NN9pAADwLmZ2jgwZMkTz5s1Tdna2unXrpi5dumjmzJnKzMyUc07VqlXTFVdc4TsmislycxQ+OlD6dZ2C6+6Sq72H70gAAECSFeQrHP+8VLe+XKvjfMcBACAmxEw50rt373987LjjeMGOR1aQr/CJe6VlixX0vFWu0b6+IwEAgN/YR29Lq1cq6NVfLuCAdAAApBi/rQbxx8JQ9uyj0rxZchdeJXfAIb4jAQCA39jmHNmksdJ+zaX/O9h3HAAAYgblCEqUvfqs7KuP5E4/X8GRbXf8GwAAQNTY2+OljVkKOl8k55zvOAAAxAzKEZSY8N3XZO+9LnfsyXInneU7DgAA+BPbsFY25TW5Q1vL1d/bdxwAAGIK5QhKRPjlR7JXnpFatJI75zLejQIAIMbYxNFSYSh3+vm+owAAEHMoR7Db7LtvZM8+IjVppuA/13G4GwAAMcZWLJV9OkWuzYly1Wv5jgMAQMyhHMFusaWLFQ67R6pVV0HPvnKppX1HAgAAfxOOf14qU0bu5LN9RwEAICZRjmCX2ZpfFD46QCqboeCa/nLp5XxHAgAAf2M/zJO++UqufSe58hV8xwEAICZRjmCX2MYshY/cIeXnK+h9h1yVar4jAQCAvzEzhf99TqpURa5tR99xAACIWZQj2GmWt0Xh43dJa1Yp6HmrXJ09fUcCAADbMutLadF8uVPPlUtL850GAICYRTmCnWJhocKnBkuLFyi4rI9ck//zHQkAAGyDFRYWnTVSq57ckW19xwEAIKZRjqDYzEw2+ilp1hdyZ18u16KV70gAAOBf2OfvSyuXKzjjfLlSPEkOAIDtoRxBsdnb/5VNfVOu/RkKjj/FdxwAAPAvLG+LbOJoqUET6aAjfMcBACDmUY6gWMIvPpSNf17u0NZynS7yHQcAAGyHfThZ2rBWQeeL5ZzzHQcAgJhHOYIdsu++kT33mLTP/nIX95IL+LIBACBW2aaNsjdflZq1kNunme84AADEBb7LxXbZ8h8VPjFIqlVXQY9b5FJTfUcCAADbYW//V9q8SUGnC31HAQAgblCO4F/ZujUKHxkgpZVVcE0/ufRyviMBAIDtsPVrZe+/UXQb7B4NfMcBACBuUI5gmyxnk8JHB0i5OQp69ZOrUt13JAAAsAM2aYwUhnIdu/qOAgBAXKEcwT9YQX7RrTQrlyvo0VeuHu88AQAQ62zlz7JP35M7poNc9Vq+4wAAEFcoR/AXZiYb9Zg0f7bchVfL7dfcdyQAAFAMNvFlKbW03Mln+Y4CAEDcoRzBX9jrL8m+mCrXsauCVsf5jgMAAIrBflokm/aJXNvT5CpU9h0HAIC4QzmCrcJP3pVNHid31AlyJ3fxHQcAABRT+NoLUkZ5uXZn+I4CAEBcohyBJMnmzpC9OEz6v4PkunaXc853JAAAUAy2YK40d6bciZ3l0jN8xwEAIC5RjkC2dJHC4fdLdesr6HaTXEqK70gAAKAYzEzhhOelSlXkjj3ZdxwAAOIW5UiSs3WrFT46UMrIUHBNP7ky6b4jAQCA4po9TVo0X+7Uc+RKp/lOAwBA3KIcSWKWs0nho3dKebkKrukvV6mq70gAAKCYLCxUOOEFqUYduVZtfccBACCuUY4kKSsoUDjiPmnlcgXdbparW993JAAAsBPsq0+kn3+SO70rt8QCALCbKEeSkJkVHb46b5bcBT3lmh7oOxIAANgJVlAgm/iyVK+BXIsjfccBACDuUY4kIXvzFdlnU+ROOVvBkWzDBQAg3tjnU6TVKxWccb5cwHIOAIDdxatpkgm//Ej22otyhx8rd9p5vuMAAICdZHlbZG+MlRrtK+3f0nccAAASAuVIErHvv5U994jUpJnchVfJOec7EgAA2Ek29S1pw1oFZ1zAazkAACWEciRJ2MqfFQ67R6pWU0GPvnKpqb4jAQCAnWS5ObK3XpWaHii3z/6+4wAAkDAoR5KAZWcpfOxOKQiKHtmbUc53JAAAsAtsykRpY5aC08/3HQUAgIRCOZLgLD9P4dC7pHVrFPS8Va56Ld+RAADALrBN2bJ3X5MOPFyuQRPfcQAASCiUIwnMwlD27CPSovkK/nOtXKN9fUcCAAC7yN4eL+VuVnB6V99RAABIOJQjCcxef1k27RO5ThfJtTzKdxwAALCL7Nf1sg/ekDuktVzd+r7jAACQcChHElT42fuyN8fJHXWCXIdOvuMAAIDdYG+9KhUUyJ12ru8oAAAkJMqRBGTzZ8teGCrt11yua3ce8wcAQByzdWtkH70l1+p4uZp1fMcBACAhUY4kGMtcrvCJQVKN2gq63SSXkuI7EgAA2A325jjJJHfK2b6jAACQsChHEohl/1r0yN5SKQquvl0unUf2AgAQz2z1Stmn78m1bidXtYbvOAAAJCzKkQRh+fkKh90jbVjHI3sBAEgQNmmsFJSSO+ks31EAAEholCMJwMxkox6VFn4nd0lvHtkLAEACsJXLZf/7UK7NiXKVqvqOAwBAQqMcSQA2aazsy4/kTj9fwSE8shcAgERgE0dLpUvLdejsOwoAAAmPciTOhV9+JJv4stwRx7LlFgCABGHLf5RN/1Tu+FPlKlTyHQcAgIRHORLHbOF3sucelZr8n9wFV/HIXgAAEkT4xmipTFm5dqf7jgIAQFKgHIlTtnpl0QGsVaop6H6LXGqq70gAAKAE2NJF0sz/ybXtKJdR3nccAACSAuVIHLKcTQofGygVFhQ9srdcBd+RAABACQknjpbSM+TanuY7CgAASYNyJM5YYaHCpx6QVq1Q0O1muVr1fEcCAAAlxJb8IH3zlVy7M+TSM3zHAQAgaVCOxBkbN1KaO1Oua3e5/Zr7jgMAAEpQOPFlKaO83PGn+I4CAEBSoRyJI+GHk2UfTJJrd7qCo9v5jgMAAEqQLZovzZ0h176TXJl033EAAEgqlCNxwr79WjbmKan5oXKdL/IdBwAAlLBw4stS+Ypyx57kOwoAAEmHciQOWOYyhSPuk+rsqeCyPnJBKd+RAABACbLvv5XmzZLr0EmuTFnfcQAASDqUIzHONmYVPZkmtbSCq25nwQQAQAIKJ74sVawsdwy7RgAA8IFyJIZZQb7CJwZJ69cq6NFXrmp135EAAEAJswVzpAVz5Dp0lktL8x0HAICkRDkSo8xM9uIT0vffyl18jVyjfX1HAgAAERBOHC1VrCLXur3vKAAAJC3KkRhl770m+2yK3ClnKzjsGN9xAABABNj82dL3c+VO7CxXml0jAAD4QjkSg+ybabJXn5NatJI79VzfcQAAQASY2W9njbBrBAAA3yhHYoz9/JPCpx6U9myk4JJr5QL+igAASEjzZ0s/zJM78Uy51NK+0wAAkNT4zjuGWPavRU+mKVNWQc9bOZQNAIAEVbRrZLRUqapc63a+4wAAkPQoR2LE1ifTZG0oKkYqV/UdCQAARMr82dLCeXInsWsEAIBYQDkSA7Y+meaHeUVPpmnQ2HckAAAQIVvPGqlUVe6oE3zHAQAAohyJCfbe6388mebQ1r7jAACASPruG2nhd3InncWuEQAAYgTliGc2Z3rRk2kO5sk0AAAkOjNT+MZodo0AABBjKEc8ssxlRU+mqVdfwaW9eTINAACJbv7s33aNnCmXmuo7DQAA+A3fjXtiG7OKnkyTkqqg521yaWV8RwIAABH0x66RKuwaAQAgxlCOeGAFBQpH3C+tX6OgR1+5qtV9RwIAAJG2YE7R4esdeEINAACxhnLEAxv7tDR/ttwFPeX23s93HAAAEAXhG2OkilXkWrfzHQUAAPwN5UiUhVPfkk19U67dGQpaHe87DgAAiAJbMFf6fq5ch07sGgEAIAZRjkSRLZgjG/OktH9Luc4X+o4DAACiJHxjtFSxslzr9r6jAACAbaAciRJbvVLh8Hul6rUVXNZHLijlOxIAAIgC+/5bacEcufad5Eqn+Y4DAAC2gXIkCiw3R+HQu6UwVHDVbXLpGb4jAQCAKAknjZEqVJJr3cF3FAAA8C8oRyLMwlDhyCHSimUKrrxRrmYd35EAAECU2MLvpO++kWt/hlwau0YAAIhVlCMRZm+MkWZ9IdflUrmmB/mOAwAAoiicPFYqV0HumBN9RwEAANtBORJBNv1T2aQxckceL3f8qb7jAACAKLIl30tzZ8q1O10urYzvOAAAYDsoRyLEli5W+OwjUqN95br2kHPOdyQAABBF4eRxUkZ5uWNP8h0FAADsAOVIBFj2rwqH3SOll1PQ/Ra51FTfkQAAQBTZ0sXSN1/JtT1Vrky67zgAAGAHKEdKmBUUFD2yN2uDgp595SpW9h0JAABEWTh5nFQ2Xe64U3xHAQAAxUA5UsJs7FPS99/KXXS13F6NfccBAABRZj//JM38XO74U+XSy/mOAwAAioFypASFH70tm/qWXPtOCg47xnccAADggU0eJ6WVlWt7mu8oAACgmChHSoh9/61s9AipWQu5Thf4jgMAADywzOWy6Z/KHXuSXEZ533EAAEAxUY6UAFu3uuickWq1FFzeRy4o5TsSAADwwN56RUotLXdCR99RAADATqAc2U2Wt0Xh0Huk/LyiA1i5txgAgKRkq1fKvvxIrnUHuQqVfMcBAAA7gXJkN5iZ7PnHpWWLFVzWR672Hr4jAQAAT+zt/0pBKbn2p/uOAgAAdhLlyG6wd18reoeoY1e55of6jgMAADyxdWtkn70vd9QJcpWq+o4DAAB2EuXILrJvv5b9d5TUopXcSWf5jgMAADyydydIMrkOnXxHAQAAu4ByZBfYqkyFTz4g1dlDwcW95JzzHQkAAHhiWetlH78jd3gbuao1fMcBAAC7gHJkJ1nuZoXD7pEkBT1vlStT1nMiAADgk737ulRQINfhTN9RAADALkrxHUCShg0bppkzZ6pixYoaPHiwJOmFF17QjBkzlJKSopo1a6pHjx7KyMjwmtPMFD73iLRimYLe/eWq1/KaBwAA+BVm/Sqb+pbcIUfJ1arrOw4AANhFMbFzpE2bNurbt+9fPnbAAQdo8ODBevDBB1W7dm1NmDDBU7o/5Ix/QZrxuVzni+SaHuQ7DgAA8Cxn8ivSls2cPwYAQJyLiXKkadOmKleu3F8+1rx5c5UqVUqS1KRJE61bt85HtK1sznRtfGmE3KGt5drxiD4AAJKd5WwqKkcOOlyubn3fcQAAwG6IidtqduSDDz5Qq1at/vXXp0yZoilTpkiS7r33XlWrVq1E//tWUKA1o59UaoPGqnzdHXJpZUr08+OfUlJSSvzvEdvGrKOLeUcPs46eWJ11pNcHOZM/VPambFXpeoVSY/DPn4hi9WstETHr6GHW0cOsoyceZ+3MzHyHkKRVq1bpvvvu23rmyO/Gjx+vRYsW6frrry/2U2FWrFhR4vls5c+qUqOG1gepJf658U/VqlXTmjVrfMdICsw6uph39DDr6InkrOvUqVNin6uk1wcWFqriymXKqrNXiX5e/Dv+v44eZh09zDp6mHX0xMva4M9i4raafzN16lTNmDFD11xzjffH5bpadVWqRm2vGQAAQOxwQSmVPqCl7xgAAKAExGw5MmvWLL3++uu66aablJaW5jsOAAAAAABIUDFx5siQIUM0b948ZWdnq1u3burSpYsmTJiggoICDRw4UJLUuHFjXXHFFZ6TAgAAAACARBMT5Ujv3r3/8bHjjjvOQxIAAAAAAJBsYva2GgAAAAAAgGigHAEAAAAAAEmNcgQAAAAAACQ1yhEAAAAAAJDUKEcAAAAAAEBSoxwBAAAAAABJjXIEAAAAAAAkNcoRAAAAAACQ1ChHAAAAAABAUqMcAQAAAAAASY1yBAAAAAAAJDXKEQAAAAAAkNQoRwAAAAAAQFKjHAEAAAAAAEmNcgQAAAAAACQ1yhEAAAAAAJDUKEcAAAAAAEBSoxwBAAAAAABJjXIEAAAAAAAkNcoRAAAAAACQ1ChHAAAAAABAUqMcAQAAAAAASY1yBAAAAAAAJDXKEQAAAAAAkNQoRwAAAAAAQFKjHAEAAAAAAEmNcgQAAAAAACQ1yhEAAAAAAJDUKEcAAAAAAEBSoxwBAAAAAABJjXIEAAAAAAAkNWdm5jsEAAAAAACAL+wc2Qk333yz7whJg1lHD7OOLuYdPcw6epJ51sn8Z/eBeUcPs44eZh09zDp64nHWlCMAAAAAACCpUY4AAAAAAICkVuqOO+64w3eIeNKwYUPfEZIGs44eZh1dzDt6mHX0JPOsk/nP7gPzjh5mHT3MOnqYdfTE26w5kBUAAAAAACQ1bqsBAAAAAABJLcV3gFg0a9YsPfvsswrDUMcff7xOP/30v/x6fn6+Hn/8cS1evFjly5dX7969VaNGDU9p49uOZj1p0iS9//77KlWqlCpUqKDu3burevXqntLGtx3N+ndffPGFHnroIQ0aNEiNGjWKcsrEUJxZf/7553rllVfknFP9+vXVq1cvD0kTw47mvWbNGg0dOlSbNm1SGIY677zzdPDBB3tKG7+GDRummTNnqmLFiho8ePA/ft3M9Oyzz+rrr79WWlqaevToEXfbabeHtUH0sDaILtYH0cP6IHpYG0RPQq0PDH9RWFhoV111la1cudLy8/Pt+uuvt2XLlv3lmrfffttGjBhhZmaffvqpPfTQQz6ixr3izHrOnDmWm5trZmbvvPMOs95FxZm1mVlOTo7169fP+vbtawsXLvSQNP4VZ9YrVqywG264wbKzs83MbMOGDT6iJoTizHv48OH2zjvvmJnZsmXLrEePHj6ixr1vv/3WFi1aZNddd902f33GjBl29913WxiGtmDBArvllluinDByWBtED2uD6GJ9ED2sD6KHtUF0JdL6gNtq/mbhwoWqVauWatasqZSUFLVq1UrTpk37yzXTp09XmzZtJEmHH3645s6dK+Polp1WnFk3a9ZMaWlpkqTGjRtr3bp1PqLGveLMWpLGjh2r0047TampqR5SJobizPr9999X+/btVa5cOUlSxYoVfURNCMWZt3NOOTk5kqScnBxVrlzZR9S417Rp061fs9syffp0tW7dWs45NWnSRJs2bdL69eujmDByWBtED2uD6GJ9ED2sD6KHtUF0JdL6gHLkb9atW6eqVatu/XnVqlX/8aL752tKlSql9PR0ZWdnRzVnIijOrP/sgw8+0IEHHhiNaAmnOLNesmSJ1qxZoxYtWkQ7XkIpzqxXrFihzMxM3X777br11ls1a9asaMdMGMWZ91lnnaVPPvlE3bp106BBg3TppZdGO2ZSWLdunapVq7b15zv6Nz2esDaIHtYG0cX6IHpYH0QPa4PYEk/rA8qRv9nWuzzOuZ2+Bju2M3P8+OOPtXjxYp122mmRjpWQdjTrMAw1atQoXXjhhdGMlZCK83UdhqEyMzPVv39/9erVS8OHD9emTZuiFTGhFGfen332mdq0aaPhw4frlltu0WOPPaYwDKMVMWkk8msja4PoYW0QXawPoof1QfSwNogt8fT6SDnyN1WrVtXatWu3/nzt2rX/2Gb152sKCwuVk5Oz3a1E2LbizFqSZs+erQkTJujGG29kO+cu2tGsc3NztWzZMg0YMEA9e/bUDz/8oPvvv1+LFi3yETeuFefrukqVKjrkkEOUkpKiGjVqqE6dOsrMzIx21IRQnHl/8MEHOuKIIyRJTZo0UX5+Pu/oR0DVqlW1Zs2arT//t3/T4xFrg+hhbRBdrA+ih/VB9LA2iC3xtD6gHPmbRo0aKTMzU6tWrVJBQYE+//xztWzZ8i/XtGjRQlOnTpVUdHL3//3f/8Vs+xXLijPrJUuW6KmnntKNN97IfZe7YUezTk9P18iRIzV06FANHTpUjRs31o033shp9LugOF/Xhx56qObOnStJysrKUmZmpmrWrOkjbtwrzryrVau2dd7Lly9Xfn6+KlSo4CNuQmvZsqU+/vhjmZm+//57paenx+ziZ2exNoge1gbRxfogelgfRA9rg9gST+sDZ5wW9g8zZ87UqFGjFIahjj32WHXq1Eljx45Vo0aN1LJlS+Xl5enxxx/XkiVLVK5cOfXu3Zt/uHbRjmY9cOBALV26VJUqVZJU9A/ZTTfd5Dl1fNrRrP/sjjvu0AUXXMDiZxftaNZmpueff16zZs1SEATq1KmTjjzySN+x49aO5r18+XKNGDFCubm5kqTzzz9fzZs395w6/gwZMkTz5s1Tdna2KlasqC5duqigoECS1K5dO5mZRo4cqW+++UalS5dWjx49EurfENYG0cPaILpYH0QP64PoYW0QPYm0PqAcAQAAAAAASY3bagAAAAAAQFKjHAEAAAAAAEmNcgQAAAAAACQ1yhEAAAAAAJDUKEcAAAAAAEBSoxwBAAAAAABJjXIEQExbsGCBunTpoi5duui7777b4fX9+/dXly5ddPfdd0chHQAA8IH1AYCSRjkCIKYtWbJEkuScU4MGDbZ7bRiGW6/f0bUAACB+sT4AUNIoRwDEtMWLF0uS6tSpozJlymz32hUrVig3N1eS1LBhw4hnAwAAfrA+AFDSKEcAxLTf3+kpzmLm94VSca8HAADxifUBgJJGOQIgZuXl5Wn58uWSdm7xU758edWoUSOi2QAAgB+sDwBEAuUIgJi1dOlSFRYWStq5xQ/vCgEAkLhYHwCIBMoRADHr98VMcQ9b+/HHHyVx2BoAAImM9QGASKAcARCzOGwNAAD8HesDAJFAOQIgZnHYGgAA+DvWBwAigXIEQEwqKCjQsmXLJEmNGjXa4fUctgYAQOJjfQAgUihHAMSkpUuXqqCgQFLx3ulZtGhRsa8FAADxifUBgEihHAEQk3766aetP95zzz23e21eXt7Wd4aaNGkS0VwAAMAf1gcAIoVyBEBMWrVqlSQpJSXl/9u5Q91EwigMw9+EFU11SUhoLwCFwFWMqOMCkNxObwWJqcDXNSFoFFUYDE1GgWDFJqvaqp0O2f951Igjjjx5JzO5vb39dnaz2eR8PidJxuNx67sBAN1wHwBtEUeAq1RVVZI/3xZ/fHx8O/vy8pIkGQ6H3gwBwH/MfQC0RRwBrtLDw8Pf59fX1y/nlstlttttkmQ2m7W+FwDQHfcB0JZfXS8A8JnJZJJ+v5/D4ZDFYpGmaVLXde7u7nI+n7Pb7bJarfL29pYkqes6j4+PHW8NALTJfQC0pbpcLpeulwD4zPv7e56fn3M8Hr+cqaoq0+k08/k8vV7vB7cDALrgPgDaII4AV61pmqxWq6zX6+z3+5xOp9zc3GQwGGQ0GuXp6Sn39/ddrwkA/CD3AfCviSMAAABA0fyQFQAAACiaOAIAAAAUTRwBAAAAiiaOAAAAAEUTRwAAAICiiSMAAABA0cQRAAAAoGjiCAAAAFA0cQQAAAAomjgCAAAAFE0cAQAAAIr2G/5MxfeNdR+RAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 1080x1080 with 4 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(2, 2, sharex=True, sharey=True, figsize=(15, 15))\n",
    "\n",
    "us = np.linspace(0, 1, 100)\n",
    "\n",
    "axes[0,0].plot(us, _equilibrium_total_offspring(us, 10, 8, 6, 1, 0))\n",
    "axes[0,0].set_title(r\"$(i)\\ T + S < 2P < R + P < 2R$\", fontsize=25)\n",
    "axes[0,0].set_ylabel(r\"$N$\", rotation=\"horizontal\", fontsize=25)\n",
    "\n",
    "axes[0,1].plot(us, _equilibrium_total_offspring(us, 10, 8, 6, 3, 0))\n",
    "axes[0,1].set_title(r\"$(ii)\\ 2P < T + S < R + P < 2R$\", fontsize=25)\n",
    "\n",
    "axes[1,0].plot(us, _equilibrium_total_offspring(us, 10, 8, 6, 5, 0))\n",
    "axes[1,0].set_title(r\"$(iii)\\ 2P < R + P < T + S < 2R$\", fontsize=25)\n",
    "axes[1,0].set_xlabel(r\"$U$\", fontsize=25)\n",
    "axes[1,0].set_ylabel(r\"$N$\", rotation=\"horizontal\", fontsize=25)\n",
    "\n",
    "axes[1,1].plot(us, _equilibrium_total_offspring(us, 14, 8, 6, 3, 0))\n",
    "axes[1,1].set_title(r\"$(iv)\\ 2P < R + P < 2R < T + S$\", fontsize=25)\n",
    "axes[1,1].set_xlabel(r\"$U$\", fontsize=25)\n",
    "\n",
    "fig.tight_layout()\n",
    "fig.subplots_adjust(top=0.925)\n",
    "fig.suptitle(\"Total offspring, $N$, as a function of the selection probability, $U$\", x=0.5, y=1.0, fontsize=35)\n",
    "\n",
    "plt.savefig(\"maximizing-total-offspring-various-payoffs.jpg\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "(f1,), *_ = models.offspring_genotypes_evolution(W, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = sym.Function('N')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 172,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-x1 - 4.0*(0.5*M*epsilon - 0.5*P*epsilon + (0.25*epsilon*x1 + 0.125*x1)*N(x1) + (-0.5*P*epsilon - 0.25*R + 0.25*S + 0.5*T*epsilon)*U(x1)**2 + (-0.5*M*epsilon + 0.25*M + 1.0*P*epsilon - 0.25*S - 0.5*T*epsilon)*U(x1))/N(x1)"
      ]
     },
     "execution_count": 172,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor((sym.factor(sym.together(f1 + x1), U(x1), x1, epsilon)\n",
    "    .subs({(-0.5*M + 0.5*P + (-1.0*P + 0.5*S + 0.5*T)*U(x1) + (0.5*P + 0.5*R - 0.5*S - 0.5*T)*U(x1)**2): 0.25 * N(x1),\n",
    "           epsilon*x1*(-0.5*M + 0.5*P) + epsilon*x1*(-1.0*P + 0.5*S + 0.5*T)*U(x1) + epsilon*x1*(0.5*P + 0.5*R - 0.5*S - 0.5*T)*U(x1)**2: 0.25 * x1 * epsilon * N(x1),\n",
    "           x1*(0.25*M - 0.25*P) + x1*(0.5*P - 0.25*S - 0.25*T)*U(x1) + x1*(-0.25*P - 0.25*R + 0.25*S + 0.25*T)*U(x1)**2: 0.125 * x1 * N(x1)})), N(x1), U(x1)) - x1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 167,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.125"
      ]
     },
     "execution_count": 167,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 / 8"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The locus of potential equilibria (LPE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - x_{1} + \\frac{x_{1} \\left(UG_{star} \\left(1 - \\epsilon\\right) \\left(2 UG_{star} \\left(- M + R\\right) + 2 \\left(1 - UG_{star}\\right) \\left(- M + S\\right)\\right) + 0.5 \\left(1 - UG_{star}\\right) \\left(2 UG_{star} \\left(- M + T\\right) + 2 \\left(1 - UG_{star}\\right) \\left(- M + P\\right)\\right)\\right) + \\left(1 - x_{1}\\right) \\left(0.5 UG_{star} \\left(2 UG_{star} \\left(- M + R\\right) + 2 \\left(1 - UG_{star}\\right) \\left(- M + S\\right)\\right) + \\epsilon \\left(1 - UG_{star}\\right) \\left(2 UG_{star} \\left(- M + T\\right) + 2 \\left(1 - UG_{star}\\right) \\left(- M + P\\right)\\right)\\right)}{x_{1} \\left(UG_{star} \\epsilon \\left(2 UG_{star} \\left(- M + R\\right) + 2 \\left(1 - UG_{star}\\right) \\left(- M + S\\right)\\right) + 0.5 \\left(1 - UG_{star}\\right) \\left(2 UG_{star} \\left(- M + T\\right) + 2 \\left(1 - UG_{star}\\right) \\left(- M + P\\right)\\right)\\right) + x_{1} \\left(UG_{star} \\left(1 - \\epsilon\\right) \\left(2 UG_{star} \\left(- M + R\\right) + 2 \\left(1 - UG_{star}\\right) \\left(- M + S\\right)\\right) + 0.5 \\left(1 - UG_{star}\\right) \\left(2 UG_{star} \\left(- M + T\\right) + 2 \\left(1 - UG_{star}\\right) \\left(- M + P\\right)\\right)\\right) + \\left(1 - x_{1}\\right) \\left(0.5 UG_{star} \\left(2 UG_{star} \\left(- M + R\\right) + 2 \\left(1 - UG_{star}\\right) \\left(- M + S\\right)\\right) + \\epsilon \\left(1 - UG_{star}\\right) \\left(2 UG_{star} \\left(- M + T\\right) + 2 \\left(1 - UG_{star}\\right) \\left(- M + P\\right)\\right)\\right) + \\left(1 - x_{1}\\right) \\left(0.5 UG_{star} \\left(2 UG_{star} \\left(- M + R\\right) + 2 \\left(1 - UG_{star}\\right) \\left(- M + S\\right)\\right) + \\left(1 - UG_{star}\\right) \\left(1 - \\epsilon\\right) \\left(2 UG_{star} \\left(- M + T\\right) + 2 \\left(1 - UG_{star}\\right) \\left(- M + P\\right)\\right)\\right)}$"
      ],
      "text/plain": [
       "-x1 + (x1*(UG_star*(1 - epsilon)*(2*UG_star*(-M + R) + 2*(1 - UG_star)*(-M + S)) + 0.5*(1 - UG_star)*(2*UG_star*(-M + T) + 2*(1 - UG_star)*(-M + P))) + (1 - x1)*(0.5*UG_star*(2*UG_star*(-M + R) + 2*(1 - UG_star)*(-M + S)) + epsilon*(1 - UG_star)*(2*UG_star*(-M + T) + 2*(1 - UG_star)*(-M + P))))/(x1*(UG_star*epsilon*(2*UG_star*(-M + R) + 2*(1 - UG_star)*(-M + S)) + 0.5*(1 - UG_star)*(2*UG_star*(-M + T) + 2*(1 - UG_star)*(-M + P))) + x1*(UG_star*(1 - epsilon)*(2*UG_star*(-M + R) + 2*(1 - UG_star)*(-M + S)) + 0.5*(1 - UG_star)*(2*UG_star*(-M + T) + 2*(1 - UG_star)*(-M + P))) + (1 - x1)*(0.5*UG_star*(2*UG_star*(-M + R) + 2*(1 - UG_star)*(-M + S)) + epsilon*(1 - UG_star)*(2*UG_star*(-M + T) + 2*(1 - UG_star)*(-M + P))) + (1 - x1)*(0.5*UG_star*(2*UG_star*(-M + R) + 2*(1 - UG_star)*(-M + S)) + (1 - UG_star)*(1 - epsilon)*(2*UG_star*(-M + T) + 2*(1 - UG_star)*(-M + P))))"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 86,
   "metadata": {},
   "outputs": [],
   "source": [
    "_x1_star, = sym.solve(f1.subs({UGA(x1 + x3): UG_star, x2: 1 - x1, x3: 0}), x1)\n",
    "_x1_star = sym.simplify(sym.factor(x1_star, UG_star).subs({UG_star: 1}))\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{1}{2.0 \\epsilon + 1.0}$"
      ],
      "text/plain": [
       "1/(2.0*epsilon + 1.0)"
      ]
     },
     "execution_count": 87,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_x1_star"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ L(x_1^*; U^*) = x_1^* - \\Bigg(\\frac{1}{(1 + 2\\epsilon)}\\frac{(U^{*2}\\epsilon(2P - 2T) + U^{*2}(R - S) + U^*\\epsilon(2M - 4P + 2T) + U^*(-M + S) + \\epsilon(-2M + 2P))}{((-M + P + U^{*2}(P + R - S - T) + U^*(-2P + S + T)))}\\Bigg) $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Setting the equation of motion equal to zero, rearranging the resulting equation yields the locus of potential equilibria (LPE). The LPE represents, for each possible $x_1$, the value that $U_{GA}$ would have to take in order for that $x_1$ to be an equilbrium. \n",
    "\n",
    "\\begin{align}\n",
    "  L(x^*) \\equiv x_1^* - 2\\left(\\frac{\\big(R - S\\big)U_{GA}^{*2} + \\big(S - M\\big)U_{GA}^*}{N(U_{GA}^*)}\\right) = 0\\\\\n",
    "\\end{align}\n",
    "\n",
    "We already know that $x_1^*=0, U_{GA}^*=0$ and $x_1^*=1, U_{GA}^*=1$ are equilibria, so it follows that $L(0)=0$ and $L(1)=0$.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Differentiate $x_1^*$ with respect to $U_{GA}^*$ yields an expression that will be useful for assessing stability and invadability of an equilibrium."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "dx1star_dUGA_star = sym.factor(sym.cancel(sym.diff(x1_star, UGA_star, 1)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.5*(1.0*epsilon - 0.5)*(0.5*M**2 + 0.5*M*P*UGA_star**2 - 0.5*M*P + 0.5*M*R*UGA_star**2 - 1.0*M*R*UGA_star - 0.5*M*S*UGA_star**2 + 1.0*M*S*UGA_star - 0.5*M*S - 0.5*M*T*UGA_star**2 - 1.0*P*R*UGA_star**2 + 1.0*P*R*UGA_star + 0.5*P*S*UGA_star**2 - 1.0*P*S*UGA_star + 0.5*P*S + 0.5*R*T*UGA_star**2)/((1.0*epsilon + 0.5)*(0.5*M - 0.5*P*UGA_star**2 + 1.0*P*UGA_star - 0.5*P - 0.5*R*UGA_star**2 + 0.5*S*UGA_star**2 - 0.5*S*UGA_star + 0.5*T*UGA_star**2 - 0.5*T*UGA_star)**2)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dx1star_dUGA_star"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Substituting the equilibrium value for $U_{GA}^*$ that maximizes total offspring, $\\bar{U}_{GA}^*$, into the expression for $x_1^*$ we can find the equilibrium value of $x_1^*$ that maximizes total offspring.\n",
    "\n",
    "\\begin{align}\n",
    "   % \\frac{(P - \\frac{1}{2}S - \\frac{1}{2}T)(PR + \\frac{1}{2}RS - \\frac{1}{2}RT - \\frac{1}{2}S^2 - \\frac{1}{2}ST)}{((\\frac{1}{2}P + \\frac{1}{2}R - \\frac{1}{2}S - \\frac{1}{2}T)(2PR - \\frac{1}{2}S^2 - ST - \\frac{1}{2}T^2))} \\\\\n",
    "   % \\frac{\\big(2P - (T + S)\\big)(PR + \\frac{1}{2}RS - \\frac{1}{2}RT - \\frac{1}{2}S^2 - \\frac{1}{2}ST)}{\\big((R + P) - (T + S)\\big)(2PR - \\frac{1}{2}S^2 - ST - \\frac{1}{2}T^2)} \\\\\n",
    "   % \\frac{\\big(2P - (T + S)\\big)\\big(2PR + RS - RT - S^2 - ST\\big)}{\\big((R + P) - (T + S)\\big)\\big(4PR - S^2 - 2ST - T^2\\big)} \\\\\n",
    "   % \\left(\\frac{2P - (T + S)}{(R + P) - (T + S)}\\right)\\left(\\frac{2PR + RS - RT - S^2 - ST}{4PR - S^2 - 2ST - T^2}\\right) \\\\\n",
    "   % \\left(\\frac{2P - (T + S)}{(R + P) - (T + S)}\\right)\\left(\\frac{2PR + RS - RT - S^2 - ST}{4PR - (T + S)^2}\\right) \\\\\n",
    "   % \\frac{1}{2}\\left(\\frac{2P - (T + S)}{(R + P) - (T + S)}\\right)\\left(\\frac{2(2PR + RS - RT - S^2 - ST)}{4PR - (T + S)^2}\\right) \\\\\n",
    "   % \\frac{1}{2}\\left(\\frac{2P - (T + S)}{(R + P) - (T + S)}\\right)\\left(\\frac{4PR + 2RS - 2RT - 2S^2 - 2ST}{4PR - (T + S)^2}\\right) \\\\\n",
    "   % \\frac{1}{2}\\left(\\frac{2P - (T + S)}{(R + P) - (T + S)}\\right)\\left(\\frac{4PR + 2RS - 2RT -4RS + 4RS - 2S^2 - 2ST}{4PR - (T + S)^2}\\right) \\\\\n",
    "   % \\frac{1}{2}\\left(\\frac{2P - (T + S)}{(R + P) - (T + S)}\\right)\\left(\\frac{2R\\big(2P + S - T - 2S\\big) + 2S\\big(2R - S - T\\big)}{4PR - (T + S)^2}\\right) \\\\\n",
    "   % \\frac{1}{2}\\left(\\frac{2P - (T + S)}{(R + P) - (T + S)}\\right)\\left(\\frac{2R\\big(2P - (T + S)\\big) + 2S\\big(2R - (T + S)\\big)}{4PR - (T + S)^2}\\right) \\\\\n",
    "    \\bar{x}_1^* = \\left(\\frac{2R\\big(2P - (T + S)\\big) + 2S\\big(2R - (T + S)\\big)}{4PR - (T + S)^2}\\right)\\bar{U}_{GA}^*\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "optimal_x1_star = sym.factor(sym.cancel(x1_star.subs({UGA_star: optimal_UGA_star})))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5*(0.333333333333333*M*P**2 + 0.666666666666667*M*P*R*epsilon + 0.333333333333333*M*P*R - 0.333333333333333*M*P*S*epsilon - 0.5*M*P*S - 0.333333333333333*M*P*T*epsilon - 0.5*M*P*T + 0.666666666666667*M*R**2*epsilon - 1.0*M*R*S*epsilon - 0.166666666666667*M*R*S - 1.0*M*R*T*epsilon - 0.166666666666667*M*R*T + 0.333333333333333*M*S**2*epsilon + 0.166666666666667*M*S**2 + 0.666666666666667*M*S*T*epsilon + 0.333333333333333*M*S*T + 0.333333333333333*M*T**2*epsilon + 0.166666666666667*M*T**2 - 0.333333333333333*P**2*R - 0.666666666666667*P*R**2*epsilon + 0.666666666666667*P*R*S*epsilon + 0.333333333333333*P*R*T - 0.166666666666667*P*S**2*epsilon + 0.166666666666667*P*S**2 + 0.166666666666667*P*S*T + 0.166666666666667*P*T**2*epsilon + 0.0833333333333333*R*S**2 + 0.333333333333333*R*S*T*epsilon + 0.333333333333333*R*T**2*epsilon - 0.0833333333333333*R*T**2 - 0.0833333333333333*S**3 - 0.166666666666667*S**2*T*epsilon - 0.166666666666667*S**2*T - 0.333333333333333*S*T**2*epsilon - 0.0833333333333333*S*T**2 - 0.166666666666667*T**3*epsilon)/((1.0*epsilon + 0.5)*(1.0*P + 1.0*R - 1.0*S - 1.0*T)*(1.0*M*P + 1.0*M*R - 1.0*M*S - 1.0*M*T - 1.0*P*R + 0.25*S**2 + 0.5*S*T + 0.25*T**2))"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimal_x1_star"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Interactive plot of the locus of potential equilibria"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "_locus_of_potential_equilibria = sym.lambdify((UG_star, T, R, P, S),\n",
    "                                              x1_star,\n",
    "                                              modules=\"numpy\")\n",
    "\n",
    "\n",
    "def plot_locus_of_potential_equilibria(selection_function, d1, T, R, P, S):\n",
    "    assert T > R > P > S\n",
    "    us = np.linspace(0, 1, 100)\n",
    "    xs = _locus_of_potential_equilibria(us, T, R, P, S)\n",
    "\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10, 8))\n",
    "    ax.plot(xs, us, label=\"LPE\")\n",
    "    ax.set_xlabel(r\"$x_1^*$\", fontsize=25)\n",
    "    ax.set_ylabel(r\"$U*$\", rotation=\"horizontal\", fontsize=25)\n",
    "    ax.plot(us, us, 'k--', label=\"random mating\")\n",
    "    \n",
    "    # create the selection functions\n",
    "    if selection_function == \"kirkpatrick\":\n",
    "        UGA = lambda x_A: selection_functions.kirkpatrick_selection(x_A, d1)\n",
    "        ax.plot(xs, UGA(xs), label=selection_function)\n",
    "    elif selection_function == \"seger\":\n",
    "        UGA = lambda x_A: selection_functions.seger_selection(x_A, d1)\n",
    "        ax.plot(xs, UGA(xs), label=selection_function)\n",
    "    elif selection_function == \"wright\":\n",
    "        UGA = lambda x_A: selection_functions.wright_selection(x_A, d1)\n",
    "        ax.plot(xs, UGA(xs), label=selection_function)\n",
    "    elif selection_function == \"none\":\n",
    "        pass\n",
    "    else:\n",
    "        valid_funcs = (\"kirkpatrick\", \"seger\", \"wright\", \"none\")\n",
    "        msg = \"Selection_function must be one of {}, {}, or {}.\".format(*valid_funcs)\n",
    "        raise ValueError(msg)\n",
    "        \n",
    "    ax.legend()\n",
    "        \n",
    "    return fig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "746ca7427c324d3491afe6c5f0779948",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "interactive(children=(Dropdown(description='$U_{\\\\gamma(j)A}$', options=('none', 'kirkpatrick', 'seger', 'wrig…"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "# slider used to control metabolic costs\n",
    "M_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$M$\")\n",
    "\n",
    "# slider used to control which selection function is being used\n",
    "UGA_slider = widgets.Dropdown(options=[\"none\", \"kirkpatrick\", \"seger\", \"wright\"], index=0, description=r\"$U_{\\gamma(j)A}$\")\n",
    "\n",
    "# slider that controls the parameters of the selection function\n",
    "d1_slider = widgets.FloatSlider(value=1.0, min=0.0, max=10, step=0.05, description=r\"$d_1$\")\n",
    "\n",
    "# slider that controls the parameters of the selection function\n",
    "epsilon_slider = widgets.FloatSlider(value=0.0, min=0.0, max=1, step=1e-3, description=r\"$\\epsilon$\")\n",
    "\n",
    "w = widgets.interactive(plot_locus_of_potential_equilibria, selection_function=UGA_slider, d1=d1_slider,\n",
    "                        T=T_slider, R=R_slider, P=P_slider, S=S_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "w.result.savefig(\"./images/locus-potential-equilibria.jpg\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[0.0, 1.00000000000000, -(-P + S*d)/(P + R*d**2 - S*d - T*d)]"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "d = sym.symbols('d', real=True, nonnegative=True)\n",
    "sym.solve(x1 - sym.factor(sym.together(x1_star.subs({UGA_star: selection_functions.kirkpatrick_selection(x1, d), M: 0, epsilon: 0}))), x1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "def make_plot(T, R, P, S, d):\n",
    "    fig, axes = plt.subplots(1, 2, sharey=True, figsize=(15, 10))\n",
    "\n",
    "    us = np.linspace(0, 1, 100)\n",
    "\n",
    "    axes[0].plot(_equilibrium_total_offspring(us, T, R, P, S, 0), us)\n",
    "    axes[0].set_title(r\"\", fontsize=25)\n",
    "    axes[0].set_ylim(-0.05, 1.05)\n",
    "    axes[0].set_xlabel(r\"$N$\", rotation=\"horizontal\", fontsize=25)\n",
    "    axes[0].set_ylabel(r\"$U$\", rotation=\"horizontal\", fontsize=25)\n",
    "\n",
    "    xs = _locus_of_potential_equilibria(us, T, R, P, S, 0, 0)\n",
    "    axes[1].plot(xs, us, label=\"LPE\")\n",
    "    axes[1].plot(xs, selection_functions.kirkpatrick_selection(xs, d), label=\"Kirkpatrick Selection\")\n",
    "    axes[1].set_xlabel(r\"$x_1$\", fontsize=25)\n",
    "    axes[1].set_xlim(-0.05, 1.05)\n",
    "\n",
    "    equilibrium_x1 = -(-P + S*d)/(P + R*d**2 - S*d - T*d)\n",
    "    equilibrium_U = selection_functions.kirkpatrick_selection(equilibrium_x1, d)\n",
    "    equilibrium_N = _equilibrium_total_offspring(equilibrium_U, T, R, P, S, 0)\n",
    "    axes[0].axhline(equilibrium_U, color='k', linestyle=\"dashed\")\n",
    "    axes[0].vlines(equilibrium_N, -1, equilibrium_U, 'k', linestyle=\"dashed\")\n",
    "    axes[1].hlines(equilibrium_U, -1, equilibrium_x1, 'k', linestyle=\"dashed\")\n",
    "    axes[1].vlines(equilibrium_x1, -1, equilibrium_U, 'k', linestyle=\"dashed\")\n",
    "\n",
    "    # add selection functions for comparison\n",
    "    axes[1].legend()\n",
    "    return fig"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "make_plot(20, 3, 2, 1, 5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(2.0*P*epsilon*(P - 0.5*S - 0.5*T)**2 - 4.0*P*epsilon*(P - 0.5*S - 0.5*T)*(P + R - S - T) + 2.0*P*epsilon*(P + R - S - T)**2 + R*(P - 0.5*S - 0.5*T)**2 - S*(P - 0.5*S - 0.5*T)**2 + S*(P - 0.5*S - 0.5*T)*(P + R - S - T) - 2.0*T*epsilon*(P - 0.5*S - 0.5*T)**2 + 2.0*T*epsilon*(P - 0.5*S - 0.5*T)*(P + R - S - T))/((2.0*epsilon + 1.0)*(P*(P - 0.5*S - 0.5*T)**2 - 2.0*P*(P - 0.5*S - 0.5*T)*(P + R - S - T) + P*(P + R - S - T)**2 + R*(P - 0.5*S - 0.5*T)**2 - S*(P - 0.5*S - 0.5*T)**2 + S*(P - 0.5*S - 0.5*T)*(P + R - S - T) - T*(P - 0.5*S - 0.5*T)**2 + T*(P - 0.5*S - 0.5*T)*(P + R - S - T)))"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.5*(-0.333333333333333*P**2*R - 0.666666666666667*P*R**2*epsilon + 0.666666666666667*P*R*S*epsilon + 0.333333333333333*P*R*T - 0.166666666666667*P*S**2*epsilon + 0.166666666666667*P*S**2 + 0.166666666666667*P*S*T + 0.166666666666667*P*T**2*epsilon + 0.0833333333333333*R*S**2 + 0.333333333333333*R*S*T*epsilon + 0.333333333333333*R*T**2*epsilon - 0.0833333333333333*R*T**2 - 0.0833333333333333*S**3 - 0.166666666666667*S**2*T*epsilon - 0.166666666666667*S**2*T - 0.333333333333333*S*T**2*epsilon - 0.0833333333333333*S*T**2 - 0.166666666666667*T**3*epsilon)/((1.0*epsilon + 0.5)*(1.0*P + 1.0*R - 1.0*S - 1.0*T)*(-1.0*P*R + 0.25*S**2 + 0.5*S*T + 0.25*T**2))"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimal_x1_star.subs({M:0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [],
   "source": [
    "optimal_d, = sym.solve(optimal_UGA_star - selection_functions.kirkpatrick_selection(optimal_x1_star.subs({M:0}), d), d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0*(1.0*P - 0.5*S - 0.5*T)*(1.0*P**2*R*epsilon + 0.5*P*R**2 - 0.5*P*R*S - 1.0*P*R*T*epsilon - 0.5*P*S**2*epsilon + 0.125*P*S**2 - 0.5*P*S*T*epsilon - 0.125*P*T**2 - 0.25*R*S**2*epsilon - 0.25*R*S*T + 0.25*R*T**2*epsilon - 0.25*R*T**2 + 0.25*S**3*epsilon + 0.5*S**2*T*epsilon + 0.125*S**2*T + 0.25*S*T**2*epsilon + 0.25*S*T**2 + 0.125*T**3)/((1.0*R - 0.5*S - 0.5*T)*(0.5*P**2*R + 1.0*P*R**2*epsilon - 1.0*P*R*S*epsilon - 0.5*P*R*T + 0.25*P*S**2*epsilon - 0.25*P*S**2 - 0.25*P*S*T - 0.25*P*T**2*epsilon - 0.125*R*S**2 - 0.5*R*S*T*epsilon - 0.5*R*T**2*epsilon + 0.125*R*T**2 + 0.125*S**3 + 0.25*S**2*T*epsilon + 0.25*S**2*T + 0.5*S*T**2*epsilon + 0.125*S*T**2 + 0.25*T**3*epsilon))"
      ]
     },
     "execution_count": 51,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(optimal_d)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.60606060606061"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "optimal_d.subs({T: 20, R: 3, P: 2, S: 1, epsilon: 0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x720 with 2 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig = make_plot(20, 3, 2, 1, 5.60606060606061)\n",
    "fig.savefig(\"./images/interior-uninvadable-equilibrium.jpg\", format=\"jpg\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Stability\n",
    "\n",
    "Necessary and sufficient conditions for stability require that the derivative of the equation of motion with respect to $x_1$ be strictly negative when evaluated at the potential equilibrium."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "f1_jac = f1.diff(x1, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(x1*(0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 1.0*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)) + 0.5*(-x1 + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1))*(0.5*x1*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*Derivative(UGA(x1), x1) - 0.5*x1*(-2*P*Derivative(UGA(x1), x1) + 2*T*Derivative(UGA(x1), x1))*(-UGA(x1) + 1) - x1*(-0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*Derivative(UGA(x1), x1) + (-2*P*Derivative(UGA(x1), x1) + 2*T*Derivative(UGA(x1), x1))*(-0.5*UGA(x1) + 0.5) + (2.0*R*UGA(x1) + 2.0*S*(-UGA(x1) + 1))*Derivative(UGA(x1), x1) + (2.0*R*Derivative(UGA(x1), x1) - 2.0*S*Derivative(UGA(x1), x1))*UGA(x1)) - (-x1 + 1)*(-1.0*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*Derivative(UGA(x1), x1) + (-2*P*Derivative(UGA(x1), x1) + 2*T*Derivative(UGA(x1), x1))*(-1.0*UGA(x1) + 1.0) + (1.0*R*UGA(x1) + 1.0*S*(-UGA(x1) + 1))*Derivative(UGA(x1), x1) + (1.0*R*Derivative(UGA(x1), x1) - 1.0*S*Derivative(UGA(x1), x1))*UGA(x1)) - (-0.5*x1 + 0.5)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*Derivative(UGA(x1), x1) - (-0.5*x1 + 0.5)*(2*R*Derivative(UGA(x1), x1) - 2*S*Derivative(UGA(x1), x1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) - (2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-0.5*UGA(x1) + 0.5) - (-1.0*R*UGA(x1) - 1.0*S*(-UGA(x1) + 1))*UGA(x1) - 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1))/(0.5*x1*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + x1*(0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 1.0*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)) + 0.5*(-x1 + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + (-x1 + 1)*(1.0*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)))**2 - 1 + (x1*(-0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*Derivative(UGA(x1), x1) + (-2*P*Derivative(UGA(x1), x1) + 2*T*Derivative(UGA(x1), x1))*(-0.5*UGA(x1) + 0.5) + (2.0*R*UGA(x1) + 2.0*S*(-UGA(x1) + 1))*Derivative(UGA(x1), x1) + (2.0*R*Derivative(UGA(x1), x1) - 2.0*S*Derivative(UGA(x1), x1))*UGA(x1)) + (-0.5*x1 + 0.5)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*Derivative(UGA(x1), x1) + (-0.5*x1 + 0.5)*(2*R*Derivative(UGA(x1), x1) - 2*S*Derivative(UGA(x1), x1))*UGA(x1) + 0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + (-1.0*R*UGA(x1) - 1.0*S*(-UGA(x1) + 1))*UGA(x1) + 1.0*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1))/(0.5*x1*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + x1*(0.5*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 1.0*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)) + 0.5*(-x1 + 1)*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1) + (-x1 + 1)*(1.0*(2*P*(-UGA(x1) + 1) + 2*T*UGA(x1))*(-UGA(x1) + 1) + 0.5*(2*R*UGA(x1) + 2*S*(-UGA(x1) + 1))*UGA(x1)))"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f1_jac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-8.0*M**2*epsilon*U_prime(x1) - 8.0*M**2*epsilon + 4.0*M**2*U_prime(x1) - 4.0*M**2 - 8.0*M*P*epsilon*UGA(x1)**2*U_prime(x1) + 16.0*M*P*epsilon*UGA(x1)**2 - 32.0*M*P*epsilon*UGA(x1) + 8.0*M*P*epsilon*U_prime(x1) + 16.0*M*P*epsilon + 4.0*M*P*UGA(x1)**2*U_prime(x1) + 8.0*M*P*UGA(x1)**2 - 16.0*M*P*UGA(x1) - 4.0*M*P*U_prime(x1) + 8.0*M*P - 8.0*M*R*epsilon*UGA(x1)**2*U_prime(x1) + 16.0*M*R*epsilon*UGA(x1)**2 + 16.0*M*R*epsilon*UGA(x1)*U_prime(x1) + 4.0*M*R*UGA(x1)**2*U_prime(x1) + 8.0*M*R*UGA(x1)**2 - 8.0*M*R*UGA(x1)*U_prime(x1) + 8.0*M*S*epsilon*UGA(x1)**2*U_prime(x1) - 16.0*M*S*epsilon*UGA(x1)**2 - 16.0*M*S*epsilon*UGA(x1)*U_prime(x1) + 16.0*M*S*epsilon*UGA(x1) + 8.0*M*S*epsilon*U_prime(x1) - 4.0*M*S*UGA(x1)**2*U_prime(x1) - 8.0*M*S*UGA(x1)**2 + 8.0*M*S*UGA(x1)*U_prime(x1) + 8.0*M*S*UGA(x1) - 4.0*M*S*U_prime(x1) + 8.0*M*T*epsilon*UGA(x1)**2*U_prime(x1) - 16.0*M*T*epsilon*UGA(x1)**2 + 16.0*M*T*epsilon*UGA(x1) - 4.0*M*T*UGA(x1)**2*U_prime(x1) - 8.0*M*T*UGA(x1)**2 + 8.0*M*T*UGA(x1) - 8.0*P**2*epsilon*UGA(x1)**4 + 32.0*P**2*epsilon*UGA(x1)**3 - 48.0*P**2*epsilon*UGA(x1)**2 + 32.0*P**2*epsilon*UGA(x1) - 8.0*P**2*epsilon - 4.0*P**2*UGA(x1)**4 + 16.0*P**2*UGA(x1)**3 - 24.0*P**2*UGA(x1)**2 + 16.0*P**2*UGA(x1) - 4.0*P**2 - 16.0*P*R*epsilon*UGA(x1)**4 + 32.0*P*R*epsilon*UGA(x1)**3 + 16.0*P*R*epsilon*UGA(x1)**2*U_prime(x1) - 16.0*P*R*epsilon*UGA(x1)**2 - 16.0*P*R*epsilon*UGA(x1)*U_prime(x1) - 8.0*P*R*UGA(x1)**4 + 16.0*P*R*UGA(x1)**3 - 8.0*P*R*UGA(x1)**2*U_prime(x1) - 8.0*P*R*UGA(x1)**2 + 8.0*P*R*UGA(x1)*U_prime(x1) + 16.0*P*S*epsilon*UGA(x1)**4 - 48.0*P*S*epsilon*UGA(x1)**3 - 8.0*P*S*epsilon*UGA(x1)**2*U_prime(x1) + 48.0*P*S*epsilon*UGA(x1)**2 + 16.0*P*S*epsilon*UGA(x1)*U_prime(x1) - 16.0*P*S*epsilon*UGA(x1) - 8.0*P*S*epsilon*U_prime(x1) + 8.0*P*S*UGA(x1)**4 - 24.0*P*S*UGA(x1)**3 + 4.0*P*S*UGA(x1)**2*U_prime(x1) + 24.0*P*S*UGA(x1)**2 - 8.0*P*S*UGA(x1)*U_prime(x1) - 8.0*P*S*UGA(x1) + 4.0*P*S*U_prime(x1) + 16.0*P*T*epsilon*UGA(x1)**4 - 48.0*P*T*epsilon*UGA(x1)**3 + 48.0*P*T*epsilon*UGA(x1)**2 - 16.0*P*T*epsilon*UGA(x1) + 8.0*P*T*UGA(x1)**4 - 24.0*P*T*UGA(x1)**3 + 24.0*P*T*UGA(x1)**2 - 8.0*P*T*UGA(x1) - 8.0*R**2*epsilon*UGA(x1)**4 - 4.0*R**2*UGA(x1)**4 + 16.0*R*S*epsilon*UGA(x1)**4 - 16.0*R*S*epsilon*UGA(x1)**3 + 8.0*R*S*UGA(x1)**4 - 8.0*R*S*UGA(x1)**3 + 16.0*R*T*epsilon*UGA(x1)**4 - 16.0*R*T*epsilon*UGA(x1)**3 - 8.0*R*T*epsilon*UGA(x1)**2*U_prime(x1) + 8.0*R*T*UGA(x1)**4 - 8.0*R*T*UGA(x1)**3 + 4.0*R*T*UGA(x1)**2*U_prime(x1) - 8.0*S**2*epsilon*UGA(x1)**4 + 16.0*S**2*epsilon*UGA(x1)**3 - 8.0*S**2*epsilon*UGA(x1)**2 - 4.0*S**2*UGA(x1)**4 + 8.0*S**2*UGA(x1)**3 - 4.0*S**2*UGA(x1)**2 - 16.0*S*T*epsilon*UGA(x1)**4 + 32.0*S*T*epsilon*UGA(x1)**3 - 16.0*S*T*epsilon*UGA(x1)**2 - 8.0*S*T*UGA(x1)**4 + 16.0*S*T*UGA(x1)**3 - 8.0*S*T*UGA(x1)**2 - 8.0*T**2*epsilon*UGA(x1)**4 + 16.0*T**2*epsilon*UGA(x1)**3 - 8.0*T**2*epsilon*UGA(x1)**2 - 4.0*T**2*UGA(x1)**4 + 8.0*T**2*UGA(x1)**3 - 4.0*T**2*UGA(x1)**2)/(8.0*M**2 - 16.0*M*P*UGA(x1)**2 + 32.0*M*P*UGA(x1) - 16.0*M*P - 16.0*M*R*UGA(x1)**2 + 16.0*M*S*UGA(x1)**2 - 16.0*M*S*UGA(x1) + 16.0*M*T*UGA(x1)**2 - 16.0*M*T*UGA(x1) + 8.0*P**2*UGA(x1)**4 - 32.0*P**2*UGA(x1)**3 + 48.0*P**2*UGA(x1)**2 - 32.0*P**2*UGA(x1) + 8.0*P**2 + 16.0*P*R*UGA(x1)**4 - 32.0*P*R*UGA(x1)**3 + 16.0*P*R*UGA(x1)**2 - 16.0*P*S*UGA(x1)**4 + 48.0*P*S*UGA(x1)**3 - 48.0*P*S*UGA(x1)**2 + 16.0*P*S*UGA(x1) - 16.0*P*T*UGA(x1)**4 + 48.0*P*T*UGA(x1)**3 - 48.0*P*T*UGA(x1)**2 + 16.0*P*T*UGA(x1) + 8.0*R**2*UGA(x1)**4 - 16.0*R*S*UGA(x1)**4 + 16.0*R*S*UGA(x1)**3 - 16.0*R*T*UGA(x1)**4 + 16.0*R*T*UGA(x1)**3 + 8.0*S**2*UGA(x1)**4 - 16.0*S**2*UGA(x1)**3 + 8.0*S**2*UGA(x1)**2 + 16.0*S*T*UGA(x1)**4 - 32.0*S*T*UGA(x1)**3 + 16.0*S*T*UGA(x1)**2 + 8.0*T**2*UGA(x1)**4 - 16.0*T**2*UGA(x1)**3 + 8.0*T**2*UGA(x1)**2)"
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.powsimp(sym.cancel(f_jac))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Equilibrium with $x_1^*=0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.500000000000000"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.simplify(f_jac.subs({x1: 0}))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Assuming that $M < S < P$ then we have the following stability condition for the equilibrium at $x_1^*=0$.\n",
    "\n",
    "$$ U_{GA}'(0) < \\frac{P-M}{S-M} $$\n",
    "\n",
    "Note that if $M \\ge S$, then the equilibrium at $x_1^*=0$ is always stable."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Equilibrium with $x_1^* = 1$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5*(R - T)/(M - R)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.simplify(f_jac.subs({x1: 1}))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ U_{GA}'(1) < \\frac{R - M}{T - M} $$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Equilibrium with $0 < x^* < 1$\n",
    "\n",
    "Sufficient conditions to guarantee the existence of a stable, interior equilibrium are that the stability conditions for both corner equilibria are violated. Necessary and sufficient conditions are more difficult to state..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "UGA_prime_star = sym.symbols(\"UGA_prime_star\")\n",
    "evaluated_f_jac = f_jac.subs({sym.Derivative(UGA(x1), x1): UGA_prime_star, UGA(x1): UGA_star, x1: x1_star})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Stability of the equilibrium involves determining whether the following function of $U_{GA}^*$, $U_{GA}^{'*}$ and payoffs $T,R,P,S$ is negative! "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.5*UGA_star*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1))*(-1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 1) + 1.0*(1.0*UGA_star*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1)) + 0.5*(-UGA_star + 1)*(2*UGA_star*(-M + T) + 2*(-M + P)*(-UGA_star + 1)))*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))*(-0.5*UGA_star*(2*(-M + R)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 2*(-M + S)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)))*(-1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 1) - 0.5*(-UGA_star + 1)*(-2*(-M + P)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 2*(-M + T)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)))*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 0.5*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1))*(-1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 1)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 0.5*(2*UGA_star*(-M + T) + 2*(-M + P)*(-UGA_star + 1))*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - (-1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 1)*(0.5*UGA_star*(2*(-M + R)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 2*(-M + S)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))) + 1.0*(-UGA_star + 1)*(-2*(-M + P)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 2*(-M + T)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))) + 0.5*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1))*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 1.0*(2*UGA_star*(-M + T) + 2*(-M + P)*(-UGA_star + 1))*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))) - 1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)*(1.0*UGA_star*(2*(-M + R)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 2*(-M + S)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))) + 0.5*(-UGA_star + 1)*(-2*(-M + P)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 2*(-M + T)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))) + 1.0*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1))*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 0.5*(2*UGA_star*(-M + T) + 2*(-M + P)*(-UGA_star + 1))*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)))/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))/(0.5*UGA_star*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1))*(-1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 1) + 0.5*(-UGA_star + 1)*(2*UGA_star*(-M + T) + 2*(-M + P)*(-UGA_star + 1))*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + (0.5*UGA_star*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1)) + 1.0*(-UGA_star + 1)*(2*UGA_star*(-M + T) + 2*(-M + P)*(-UGA_star + 1)))*(-1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 1) + 1.0*(1.0*UGA_star*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1)) + 0.5*(-UGA_star + 1)*(2*UGA_star*(-M + T) + 2*(-M + P)*(-UGA_star + 1)))*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))**2 - 1 + (0.5*UGA_star*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1)) + 0.5*UGA_star*(2*(-M + R)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 2*(-M + S)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)))*(-1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 1) + 0.5*(-UGA_star + 1)*(2*UGA_star*(-M + T) + 2*(-M + P)*(-UGA_star + 1)) + 0.5*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1))*(-1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 1)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)*(1.0*UGA_star*(2*(-M + R)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 2*(-M + S)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))) + 0.5*(-UGA_star + 1)*(-2*(-M + P)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 2*(-M + T)*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))) + 1.0*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1))*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 0.5*(2*UGA_star*(-M + T) + 2*(-M + P)*(-UGA_star + 1))*U_prime(1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)))/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))/(0.5*UGA_star*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1))*(-1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 1) + 0.5*(-UGA_star + 1)*(2*UGA_star*(-M + T) + 2*(-M + P)*(-UGA_star + 1))*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + (0.5*UGA_star*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1)) + 1.0*(-UGA_star + 1)*(2*UGA_star*(-M + T) + 2*(-M + P)*(-UGA_star + 1)))*(-1.0*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 1) + 1.0*(1.0*UGA_star*(2*UGA_star*(-M + R) + 2*(-M + S)*(-UGA_star + 1)) + 0.5*(-UGA_star + 1)*(2*UGA_star*(-M + T) + 2*(-M + P)*(-UGA_star + 1)))*(2.0*M*UGA_star - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star)/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star))"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "evaluated_f_jac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5*(1.0*M**2*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 1.0*M**2 + 1.0*M*P*UGA_star**2*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 2.0*M*P*UGA_star**2 - 4.0*M*P*UGA_star - 1.0*M*P*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 2.0*M*P + 1.0*M*R*UGA_star**2*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 2.0*M*R*UGA_star**2 - 2.0*M*R*UGA_star*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 1.0*M*S*UGA_star**2*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 2.0*M*S*UGA_star**2 + 2.0*M*S*UGA_star*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 2.0*M*S*UGA_star - 1.0*M*S*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 1.0*M*T*UGA_star**2*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 2.0*M*T*UGA_star**2 + 2.0*M*T*UGA_star - 1.0*P**2*UGA_star**4 + 4.0*P**2*UGA_star**3 - 6.0*P**2*UGA_star**2 + 4.0*P**2*UGA_star - 1.0*P**2 - 2.0*P*R*UGA_star**4 + 4.0*P*R*UGA_star**3 - 2.0*P*R*UGA_star**2*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 2.0*P*R*UGA_star**2 + 2.0*P*R*UGA_star*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 2.0*P*S*UGA_star**4 - 6.0*P*S*UGA_star**3 + 1.0*P*S*UGA_star**2*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 6.0*P*S*UGA_star**2 - 2.0*P*S*UGA_star*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 2.0*P*S*UGA_star + 1.0*P*S*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) + 2.0*P*T*UGA_star**4 - 6.0*P*T*UGA_star**3 + 6.0*P*T*UGA_star**2 - 2.0*P*T*UGA_star - 1.0*R**2*UGA_star**4 + 2.0*R*S*UGA_star**4 - 2.0*R*S*UGA_star**3 + 2.0*R*T*UGA_star**4 - 2.0*R*T*UGA_star**3 + 1.0*R*T*UGA_star**2*U_prime(2.0*M*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*R*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) + 2.0*S*UGA_star**2/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star) - 2.0*S*UGA_star/(2.0*M - 2.0*P*UGA_star**2 + 4.0*P*UGA_star - 2.0*P - 2.0*R*UGA_star**2 + 2.0*S*UGA_star**2 - 2.0*S*UGA_star + 2.0*T*UGA_star**2 - 2.0*T*UGA_star)) - 1.0*S**2*UGA_star**4 + 2.0*S**2*UGA_star**3 - 1.0*S**2*UGA_star**2 - 2.0*S*T*UGA_star**4 + 4.0*S*T*UGA_star**3 - 2.0*S*T*UGA_star**2 - 1.0*T**2*UGA_star**4 + 2.0*T**2*UGA_star**3 - 1.0*T**2*UGA_star**2)/(1.0*M - 1.0*P*UGA_star**2 + 2.0*P*UGA_star - 1.0*P - 1.0*R*UGA_star**2 + 1.0*S*UGA_star**2 - 1.0*S*UGA_star + 1.0*T*UGA_star**2 - 1.0*T*UGA_star)**2"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(sym.cancel(sym.together(evaluated_f_jac)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "# the above expression can be reduce to a ratio of two, two-dimensional polynomial functions in U and U'...\n",
    "numerator, denominator = sym.fraction(sym.factor(sym.cancel(sym.together(evaluated_f_jac))))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Can we prove that the denominator is strictly positive? If so, then we have reduced the problem of finding the sign of the Jacobian to finding the sign of the expression in the numerator. Yes!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1.0*(1.0*M - 1.0*P*UGA_star**2 + 2.0*P*UGA_star - 1.0*P - 1.0*R*UGA_star**2 + 1.0*S*UGA_star**2 - 1.0*S*UGA_star + 1.0*T*UGA_star**2 - 1.0*T*UGA_star)**2"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(denominator)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can solve the numerator for $U'_{GA}(x^*)$.  This will allow us to write the condition for which the Jacobian will be negative as an upper bound on $U'_{GA}(x^*)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "ename": "ValueError",
     "evalue": "not enough values to unpack (expected 1, got 0)",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m                                Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-34-544b63827706>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mupper_bound\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msym\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0msolve\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mnumerator\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mUGA_prime_star\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mValueError\u001b[0m: not enough values to unpack (expected 1, got 0)"
     ]
    }
   ],
   "source": [
    "upper_bound, = sym.solve(numerator, UGA_prime_star)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sym.factor(upper_bound)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After fiddling around with the above expression we arrive at the following condition for a negative Jacobian and a stable interior equilibrium at $0 < x^* < 1$.\n",
    "\n",
    "\\begin{align}\n",
    "   % U'_{GA}(x^*) <& \\frac{(M - PU_{GA}^{*2} + 2PU_{GA}^* - P - RU_{GA}^{*2} + SU_{GA}^{*2} - SU_{GA}^* + TU_{GA}^{*2} - TU_{GA}^*)^2}{(M^2 + MPU_{GA}^{*2} - MP + MRU_{GA}^{*2} - 2MRU_{GA}^* - MSU_{GA}^{*2} + 2MSU_{GA}^* - MS - MTU_{GA}^{*2} - 2PRU_{GA}^{*2} + 2PRU_{GA}^* + PSU_{GA}^{*2} - 2PSU_{GA}^* + PS + RTU_{GA}^{*2})} \\\\\n",
    "   % U'_{GA}(x^*) <& \\frac{\\bigg(-1(-M + PU_{GA}^{*2} - 2PU_{GA}^* + P + RU_{GA}^{*2} - SU_{GA}^{*2} + SU_{GA}^* - TU_{GA}^{*2} + TU_{GA}^*)\\bigg)^2}{(M^2 + MPU_{GA}^{*2} - MP + MRU_{GA}^{*2} - 2MRU_{GA}^* - MSU_{GA}^{*2} + 2MSU_{GA}^* - MS - MTU_{GA}^{*2} - 2PRU_{GA}^{*2} + 2PRU_{GA}^* + PSU_{GA}^{*2} - 2PSU_{GA}^* + PS + RTU_{GA}^{*2})} \\\\\n",
    "   % U'_{GA}(x^*) <& \\frac{\\bigg(\\big((R + P) - (T + S)\\big)U_{GA}^{*2} + \\big((T + S) - 2P\\big)U_{GA}^* + \\big(P - M\\big)\\bigg)^2}{MRU_{GA}^{*2} + MPU_{GA}^{*2} - MSU_{GA}^{*2} - MTU_{GA}^{*2} - 2PRU_{GA}^{*2} + PSU_{GA}^{*2} + RTU_{GA}^{*2} - 2MRU_{GA}^* + 2MSU_{GA}^* + 2PRU_{GA}^* - 2PSU_{GA}^* + PS - MS - MP + M^2 }\\\\\n",
    "   % U'_{GA}(x^*) <& \\frac{\\bigg(\\big((R + P) - (T + S)\\big)U_{GA}^{*2} + \\big((T + S) - 2P\\big)U_{GA}^* + \\big(P - M\\big)\\bigg)^2}{\\bigg(M\\big((R + P) - (T + S)\\big) + \\big(TR + PS - 2RP\\big)\\bigg)U_{GA}^{*2} + 2(R - S)(P - M)U_{GA}^* + \\big(PS - M(P + S) + M^2\\big) }\\\\\n",
    "    U'_{GA}(x^*) <& \\frac{\\bigg(\\big((R + P) - (T + S)\\big)U_{GA}^{*2} + \\big((T + S) - 2P\\big)U_{GA}^* + \\big(P - M\\big)\\bigg)^2}{\\bigg(M\\big((R + P) - (T + S)\\big) + \\big(TR + PS - 2RP\\big)\\bigg)U_{GA}^{*2} + 2(R - S)(P - M)U_{GA}^* + (P - M)(S - M)}\\\\\n",
    "\\end{align}\n",
    "\n",
    "We should be able to recover the stability conditions for the corner equilibria as a special case of the above condition. Recall that $U_{GA}(0) = 0$ and $U_{GA}(1)=1$. When $x^*=0$ the condition reduces to $U'_{GA}(0) < \\frac{P - M}{S - M}$ and when $x^*=1$ the condition reduces to $U'_{GA}(1) < \\frac{R - M}{T - M}$.\n",
    "\n",
    "Can also use a graphical approach to check the correctness of the above condition.  The above condition should be the zero contour in a contour plot of the value of the Jacobian as a function of $U_{GA}$ and $U'_{GA}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "# vectorized numerical function created from our symbolic expression\n",
    "_numeric_f_jac = sym.lambdify((UGA_star, UGA_prime_star, T, R, P, S, M), evaluated_f_jac, modules=\"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "def plot_jacobian(T, R, P, S, M):\n",
    "\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(20, 10))\n",
    "    ax.set_ylabel(r\"$U_{GA}^*$\", fontsize=20, rotation=\"horizontal\")\n",
    "    ax.set_xlabel(r\"$U_{GA}^{'*}$\", fontsize=20)\n",
    "    ax.set_title(r\"Negative values of $\\frac{\\partial \\dot{x}_1}{\\partial x_1}$ indicate stability!\", fontsize=25)\n",
    "    ax.grid(\"off\")\n",
    "\n",
    "    equilibrium_selection_probs = np.linspace(0, 1, 100).reshape(-1, 1)\n",
    "    equilibrium_selection_derivs = np.linspace(0, 10, 1000).reshape(1, -1) # sensible UGA'(0) >= 1\n",
    "    Z = _numeric_f_jac(equilibrium_selection_probs, equilibrium_selection_derivs, T, R, P, S, M)\n",
    "    cax = ax.imshow(Z, origin=\"lower\", aspect=\"auto\", vmin=-0.5, vmax=1.0)\n",
    "\n",
    "    levels = np.arange(-0.25, 1.25, 0.25)\n",
    "    contours = ax.contour(Z, levels=levels, colors='w', origin='lower')\n",
    "    ax.clabel(contours, contours.levels, inline=True, fontsize=10)\n",
    "    \n",
    "    # adjust the tick labels\n",
    "    locs, _ = plt.xticks()\n",
    "    plt.xticks(locs[1:], np.linspace(0, 10, locs.size-1))\n",
    "    locs, _ = plt.yticks()\n",
    "    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mpld3.disable_notebook()  # don't need interactive plotting for this!\n",
    "\n",
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "M_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$M$\")\n",
    "\n",
    "w = widgets.interactive(plot_jacobian, T=T_slider, R=R_slider, P=P_slider, S=S_slider, M=M_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Invadability\n",
    "\n",
    "Restrict the full model to obtain a monomorphic $\\gamma$ equilibrium, add a small amount of individuals carrying the $g$ allele of the $\\gamma$ gene and simulate."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 362,
   "metadata": {},
   "outputs": [],
   "source": [
    "x2, x3 = sym.symbols(\"x2, x3\", real=True, nonnegative=True)\n",
    "x4 = 1 - x1 - x2 - x3\n",
    "x = np.array([[x1], [x2], [x3], [x4]])\n",
    "\n",
    "xA = sym.symbols(\"xA\", real=True, nonnegative=True)\n",
    "\n",
    "UGA = selection_functions.UGA\n",
    "UgA = selection_functions.UgA\n",
    "W = models.generalized_sexual_selection(x, UGA, UgA, payoff_kernel, M, 0, epsilon)\n",
    "(f1,), (f2,), (f3,), _ = models.offspring_genotypes_evolution(W, x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 200,
   "metadata": {},
   "outputs": [],
   "source": [
    "n, d = sym.fraction(f1.subs({UGA(x1 + x3): UGA(xA), UgA(x1 + x3): UgA(xA)}) + x1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 201,
   "metadata": {},
   "outputs": [],
   "source": [
    "nn, dd = sym.fraction(sym.simplify(n))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 225,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle LG \\epsilon x_{1}^{3} - LG \\epsilon x_{1}^{2} + \\epsilon x_{1}^{2} x_{2} \\left(0.5 FG + 0.5 Fg - 0.5 LG - 0.5 Lg\\right) + \\epsilon x_{1}^{2} x_{3} \\left(3.0 M \\operatorname{UGA}{\\left(xA \\right)} - 3.0 R \\operatorname{UGA}^{2}{\\left(xA \\right)} - 1.0 R \\operatorname{UgA}^{2}{\\left(xA \\right)} + 3.0 S \\operatorname{UGA}^{2}{\\left(xA \\right)} - 3.0 S \\operatorname{UGA}{\\left(xA \\right)} + 1.0 S \\operatorname{UgA}^{2}{\\left(xA \\right)} - 1.0 S \\operatorname{UgA}{\\left(xA \\right)}\\right) + \\epsilon x_{1} x_{2}^{2} \\left(- M \\operatorname{UGA}{\\left(xA \\right)} + M - P \\operatorname{UGA}^{2}{\\left(xA \\right)} + 2 P \\operatorname{UGA}{\\left(xA \\right)} + P \\operatorname{UgA}^{2}{\\left(xA \\right)} - 2 P \\operatorname{UgA}{\\left(xA \\right)} + T \\operatorname{UGA}^{2}{\\left(xA \\right)} - T \\operatorname{UGA}{\\left(xA \\right)} - T \\operatorname{UgA}^{2}{\\left(xA \\right)} + T \\operatorname{UgA}{\\left(xA \\right)}\\right) + \\epsilon x_{1} x_{2} x_{3} \\left(2 M \\operatorname{UGA}{\\left(xA \\right)} - 2 M + 2 P \\operatorname{UGA}^{2}{\\left(xA \\right)} - 4 P \\operatorname{UGA}{\\left(xA \\right)} + 2 P \\operatorname{UgA}^{2}{\\left(xA \\right)} - 4 P \\operatorname{UgA}{\\left(xA \\right)} + 4 P - 2 T \\operatorname{UGA}^{2}{\\left(xA \\right)} + 2 T \\operatorname{UGA}{\\left(xA \\right)} - 2 T \\operatorname{UgA}^{2}{\\left(xA \\right)} + 2 T \\operatorname{UgA}{\\left(xA \\right)}\\right) + \\epsilon x_{1} x_{2} \\left(- 0.5 FG - 0.5 Fg + 0.5 LG + 0.5 Lg\\right) + \\epsilon x_{1} x_{3}^{2} \\left(1.0 M \\operatorname{UGA}{\\left(xA \\right)} - 1.0 R \\operatorname{UGA}^{2}{\\left(xA \\right)} - 1.0 R \\operatorname{UgA}^{2}{\\left(xA \\right)} + 1.0 S \\operatorname{UGA}^{2}{\\left(xA \\right)} - 1.0 S \\operatorname{UGA}{\\left(xA \\right)} + 1.0 S \\operatorname{UgA}^{2}{\\left(xA \\right)} - 1.0 S \\operatorname{UgA}{\\left(xA \\right)}\\right) + \\epsilon x_{1} x_{3} \\left(- 1.0 M \\operatorname{UGA}{\\left(xA \\right)} + 1.0 R \\operatorname{UGA}^{2}{\\left(xA \\right)} + 1.0 R \\operatorname{UgA}^{2}{\\left(xA \\right)} - 1.0 S \\operatorname{UGA}^{2}{\\left(xA \\right)} + 1.0 S \\operatorname{UGA}{\\left(xA \\right)} - 1.0 S \\operatorname{UgA}^{2}{\\left(xA \\right)} + 1.0 S \\operatorname{UgA}{\\left(xA \\right)}\\right) + \\epsilon x_{2}^{2} x_{3} \\left(- M \\operatorname{UGA}{\\left(xA \\right)} + M - P \\operatorname{UGA}^{2}{\\left(xA \\right)} + 2 P \\operatorname{UGA}{\\left(xA \\right)} + P \\operatorname{UgA}^{2}{\\left(xA \\right)} - 2 P \\operatorname{UgA}{\\left(xA \\right)} + T \\operatorname{UGA}^{2}{\\left(xA \\right)} - T \\operatorname{UGA}{\\left(xA \\right)} - T \\operatorname{UgA}^{2}{\\left(xA \\right)} + T \\operatorname{UgA}{\\left(xA \\right)}\\right) + \\epsilon x_{2} x_{3}^{2} \\left(0.5 FG + 0.5 Fg - 0.5 LG - 0.5 Lg\\right) + \\epsilon x_{2} x_{3} \\left(- 0.5 FG - 0.5 Fg + 0.5 LG + 0.5 Lg\\right) + x_{1}^{3} \\left(- 1.5 M \\operatorname{UGA}{\\left(xA \\right)} - 0.5 M + 0.5 P \\operatorname{UGA}^{2}{\\left(xA \\right)} - 1.0 P \\operatorname{UGA}{\\left(xA \\right)} + 0.5 P + 2 R \\operatorname{UGA}^{2}{\\left(xA \\right)} - 0.5 R \\operatorname{UgA}^{2}{\\left(xA \\right)} - 2 S \\operatorname{UGA}^{2}{\\left(xA \\right)} + 2 S \\operatorname{UGA}{\\left(xA \\right)} + 0.5 S \\operatorname{UgA}^{2}{\\left(xA \\right)} - 0.5 S \\operatorname{UgA}{\\left(xA \\right)} - 0.5 T \\operatorname{UGA}^{2}{\\left(xA \\right)} + 0.5 T \\operatorname{UGA}{\\left(xA \\right)}\\right) + x_{1}^{2} x_{2} \\left(- 1.5 M \\operatorname{UGA}{\\left(xA \\right)} + 0.5 M - 0.5 P \\operatorname{UGA}^{2}{\\left(xA \\right)} + 1.0 P \\operatorname{UGA}{\\left(xA \\right)} - 0.5 P + R \\operatorname{UGA}^{2}{\\left(xA \\right)} - 0.5 R \\operatorname{UgA}^{2}{\\left(xA \\right)} - S \\operatorname{UGA}^{2}{\\left(xA \\right)} + S \\operatorname{UGA}{\\left(xA \\right)} + 0.5 S \\operatorname{UgA}^{2}{\\left(xA \\right)} - 0.5 S \\operatorname{UgA}{\\left(xA \\right)} + 0.5 T \\operatorname{UGA}^{2}{\\left(xA \\right)} - 0.5 T \\operatorname{UGA}{\\left(xA \\right)}\\right) + x_{1}^{2} x_{3} \\left(0.5 FG + LG\\right) + x_{1}^{2} \\left(1.5 M \\operatorname{UGA}{\\left(xA \\right)} + 0.5 M - 0.5 P \\operatorname{UGA}^{2}{\\left(xA \\right)} + 1.0 P \\operatorname{UGA}{\\left(xA \\right)} - 0.5 P - 2 R \\operatorname{UGA}^{2}{\\left(xA \\right)} + 1.0 R \\operatorname{UgA}^{2}{\\left(xA \\right)} + 2 S \\operatorname{UGA}^{2}{\\left(xA \\right)} - 2 S \\operatorname{UGA}{\\left(xA \\right)} - 1.0 S \\operatorname{UgA}^{2}{\\left(xA \\right)} + 1.0 S \\operatorname{UgA}{\\left(xA \\right)} + 0.5 T \\operatorname{UGA}^{2}{\\left(xA \\right)} - 0.5 T \\operatorname{UGA}{\\left(xA \\right)}\\right) + x_{1} x_{2} x_{3} \\left(- 2.0 M \\operatorname{UGA}{\\left(xA \\right)} + 0.5 M - 0.5 P \\operatorname{UGA}^{2}{\\left(xA \\right)} + 1.0 P \\operatorname{UGA}{\\left(xA \\right)} - 0.5 P \\operatorname{UgA}^{2}{\\left(xA \\right)} + 1.0 P \\operatorname{UgA}{\\left(xA \\right)} - 1.0 P + 1.5 R \\operatorname{UGA}^{2}{\\left(xA \\right)} - 0.5 R \\operatorname{UgA}^{2}{\\left(xA \\right)} - 1.5 S \\operatorname{UGA}^{2}{\\left(xA \\right)} + 1.5 S \\operatorname{UGA}{\\left(xA \\right)} + 0.5 S \\operatorname{UgA}^{2}{\\left(xA \\right)} - 0.5 S \\operatorname{UgA}{\\left(xA \\right)} + 0.5 T \\operatorname{UGA}^{2}{\\left(xA \\right)} - 0.5 T \\operatorname{UGA}{\\left(xA \\right)} + 0.5 T \\operatorname{UgA}^{2}{\\left(xA \\right)} - 0.5 T \\operatorname{UgA}{\\left(xA \\right)}\\right) + x_{1} x_{2} \\left(M \\operatorname{UGA}{\\left(xA \\right)} - R \\operatorname{UGA}^{2}{\\left(xA \\right)} + 0.5 R \\operatorname{UgA}^{2}{\\left(xA \\right)} + S \\operatorname{UGA}^{2}{\\left(xA \\right)} - S \\operatorname{UGA}{\\left(xA \\right)} - 0.5 S \\operatorname{UgA}^{2}{\\left(xA \\right)} + 0.5 S \\operatorname{UgA}{\\left(xA \\right)}\\right) + x_{1} x_{3}^{2} \\left(- 0.5 M \\operatorname{UGA}{\\left(xA \\right)} - 0.5 M + 0.5 P \\operatorname{UGA}^{2}{\\left(xA \\right)} - 1.0 P \\operatorname{UGA}{\\left(xA \\right)} + 0.5 P + 1.0 R \\operatorname{UGA}^{2}{\\left(xA \\right)} + 0.5 R \\operatorname{UgA}^{2}{\\left(xA \\right)} - 1.0 S \\operatorname{UGA}^{2}{\\left(xA \\right)} + 1.0 S \\operatorname{UGA}{\\left(xA \\right)} - 0.5 S \\operatorname{UgA}^{2}{\\left(xA \\right)} + 0.5 S \\operatorname{UgA}{\\left(xA \\right)} - 0.5 T \\operatorname{UGA}^{2}{\\left(xA \\right)} + 0.5 T \\operatorname{UGA}{\\left(xA \\right)}\\right) + x_{1} x_{3} \\left(0.5 M \\operatorname{UGA}{\\left(xA \\right)} + 0.5 M - 0.5 P \\operatorname{UGA}^{2}{\\left(xA \\right)} + 1.0 P \\operatorname{UGA}{\\left(xA \\right)} - 0.5 P - 1.0 R \\operatorname{UGA}^{2}{\\left(xA \\right)} + 1.0 S \\operatorname{UGA}^{2}{\\left(xA \\right)} - 1.0 S \\operatorname{UGA}{\\left(xA \\right)} + 0.5 T \\operatorname{UGA}^{2}{\\left(xA \\right)} - 0.5 T \\operatorname{UGA}{\\left(xA \\right)}\\right) + x_{1} \\left(- 0.5 R \\operatorname{UgA}^{2}{\\left(xA \\right)} + 0.5 S \\operatorname{UgA}^{2}{\\left(xA \\right)} - 0.5 S \\operatorname{UgA}{\\left(xA \\right)}\\right) + x_{2} x_{3}^{2} \\left(- 0.5 M \\operatorname{UGA}{\\left(xA \\right)} - 0.5 P \\operatorname{UgA}^{2}{\\left(xA \\right)} + 1.0 P \\operatorname{UgA}{\\left(xA \\right)} - 0.5 P + 0.5 R \\operatorname{UGA}^{2}{\\left(xA \\right)} - 0.5 S \\operatorname{UGA}^{2}{\\left(xA \\right)} + 0.5 S \\operatorname{UGA}{\\left(xA \\right)} + 0.5 T \\operatorname{UgA}^{2}{\\left(xA \\right)} - 0.5 T \\operatorname{UgA}{\\left(xA \\right)}\\right) + x_{2} x_{3} \\left(0.5 M \\operatorname{UGA}{\\left(xA \\right)} - 0.5 R \\operatorname{UGA}^{2}{\\left(xA \\right)} + 0.5 S \\operatorname{UGA}^{2}{\\left(xA \\right)} - 0.5 S \\operatorname{UGA}{\\left(xA \\right)}\\right)$"
      ],
      "text/plain": [
       "LG*epsilon*x1**3 - LG*epsilon*x1**2 + epsilon*x1**2*x2*(0.5*FG + 0.5*Fg - 0.5*LG - 0.5*Lg) + epsilon*x1**2*x3*(3.0*M*UGA(xA) - 3.0*R*UGA(xA)**2 - 1.0*R*UgA(xA)**2 + 3.0*S*UGA(xA)**2 - 3.0*S*UGA(xA) + 1.0*S*UgA(xA)**2 - 1.0*S*UgA(xA)) + epsilon*x1*x2**2*(-M*UGA(xA) + M - P*UGA(xA)**2 + 2*P*UGA(xA) + P*UgA(xA)**2 - 2*P*UgA(xA) + T*UGA(xA)**2 - T*UGA(xA) - T*UgA(xA)**2 + T*UgA(xA)) + epsilon*x1*x2*x3*(2*M*UGA(xA) - 2*M + 2*P*UGA(xA)**2 - 4*P*UGA(xA) + 2*P*UgA(xA)**2 - 4*P*UgA(xA) + 4*P - 2*T*UGA(xA)**2 + 2*T*UGA(xA) - 2*T*UgA(xA)**2 + 2*T*UgA(xA)) + epsilon*x1*x2*(-0.5*FG - 0.5*Fg + 0.5*LG + 0.5*Lg) + epsilon*x1*x3**2*(1.0*M*UGA(xA) - 1.0*R*UGA(xA)**2 - 1.0*R*UgA(xA)**2 + 1.0*S*UGA(xA)**2 - 1.0*S*UGA(xA) + 1.0*S*UgA(xA)**2 - 1.0*S*UgA(xA)) + epsilon*x1*x3*(-1.0*M*UGA(xA) + 1.0*R*UGA(xA)**2 + 1.0*R*UgA(xA)**2 - 1.0*S*UGA(xA)**2 + 1.0*S*UGA(xA) - 1.0*S*UgA(xA)**2 + 1.0*S*UgA(xA)) + epsilon*x2**2*x3*(-M*UGA(xA) + M - P*UGA(xA)**2 + 2*P*UGA(xA) + P*UgA(xA)**2 - 2*P*UgA(xA) + T*UGA(xA)**2 - T*UGA(xA) - T*UgA(xA)**2 + T*UgA(xA)) + epsilon*x2*x3**2*(0.5*FG + 0.5*Fg - 0.5*LG - 0.5*Lg) + epsilon*x2*x3*(-0.5*FG - 0.5*Fg + 0.5*LG + 0.5*Lg) + x1**3*(-1.5*M*UGA(xA) - 0.5*M + 0.5*P*UGA(xA)**2 - 1.0*P*UGA(xA) + 0.5*P + 2*R*UGA(xA)**2 - 0.5*R*UgA(xA)**2 - 2*S*UGA(xA)**2 + 2*S*UGA(xA) + 0.5*S*UgA(xA)**2 - 0.5*S*UgA(xA) - 0.5*T*UGA(xA)**2 + 0.5*T*UGA(xA)) + x1**2*x2*(-1.5*M*UGA(xA) + 0.5*M - 0.5*P*UGA(xA)**2 + 1.0*P*UGA(xA) - 0.5*P + R*UGA(xA)**2 - 0.5*R*UgA(xA)**2 - S*UGA(xA)**2 + S*UGA(xA) + 0.5*S*UgA(xA)**2 - 0.5*S*UgA(xA) + 0.5*T*UGA(xA)**2 - 0.5*T*UGA(xA)) + x1**2*x3*(0.5*FG + LG) + x1**2*(1.5*M*UGA(xA) + 0.5*M - 0.5*P*UGA(xA)**2 + 1.0*P*UGA(xA) - 0.5*P - 2*R*UGA(xA)**2 + 1.0*R*UgA(xA)**2 + 2*S*UGA(xA)**2 - 2*S*UGA(xA) - 1.0*S*UgA(xA)**2 + 1.0*S*UgA(xA) + 0.5*T*UGA(xA)**2 - 0.5*T*UGA(xA)) + x1*x2*x3*(-2.0*M*UGA(xA) + 0.5*M - 0.5*P*UGA(xA)**2 + 1.0*P*UGA(xA) - 0.5*P*UgA(xA)**2 + 1.0*P*UgA(xA) - 1.0*P + 1.5*R*UGA(xA)**2 - 0.5*R*UgA(xA)**2 - 1.5*S*UGA(xA)**2 + 1.5*S*UGA(xA) + 0.5*S*UgA(xA)**2 - 0.5*S*UgA(xA) + 0.5*T*UGA(xA)**2 - 0.5*T*UGA(xA) + 0.5*T*UgA(xA)**2 - 0.5*T*UgA(xA)) + x1*x2*(M*UGA(xA) - R*UGA(xA)**2 + 0.5*R*UgA(xA)**2 + S*UGA(xA)**2 - S*UGA(xA) - 0.5*S*UgA(xA)**2 + 0.5*S*UgA(xA)) + x1*x3**2*(-0.5*M*UGA(xA) - 0.5*M + 0.5*P*UGA(xA)**2 - 1.0*P*UGA(xA) + 0.5*P + 1.0*R*UGA(xA)**2 + 0.5*R*UgA(xA)**2 - 1.0*S*UGA(xA)**2 + 1.0*S*UGA(xA) - 0.5*S*UgA(xA)**2 + 0.5*S*UgA(xA) - 0.5*T*UGA(xA)**2 + 0.5*T*UGA(xA)) + x1*x3*(0.5*M*UGA(xA) + 0.5*M - 0.5*P*UGA(xA)**2 + 1.0*P*UGA(xA) - 0.5*P - 1.0*R*UGA(xA)**2 + 1.0*S*UGA(xA)**2 - 1.0*S*UGA(xA) + 0.5*T*UGA(xA)**2 - 0.5*T*UGA(xA)) + x1*(-0.5*R*UgA(xA)**2 + 0.5*S*UgA(xA)**2 - 0.5*S*UgA(xA)) + x2*x3**2*(-0.5*M*UGA(xA) - 0.5*P*UgA(xA)**2 + 1.0*P*UgA(xA) - 0.5*P + 0.5*R*UGA(xA)**2 - 0.5*S*UGA(xA)**2 + 0.5*S*UGA(xA) + 0.5*T*UgA(xA)**2 - 0.5*T*UgA(xA)) + x2*x3*(0.5*M*UGA(xA) - 0.5*R*UGA(xA)**2 + 0.5*S*UGA(xA)**2 - 0.5*S*UGA(xA))"
      ]
     },
     "execution_count": 225,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "LG, Lg = sym.symbols(\"LG, Lg\", real=True, nonnegative=True)\n",
    "str(sym.factor(sym.expand(nn), epsilon, x1, x2, x3)\n",
    "       .subs({2*M*UGA(xA) - 2*R*UGA(xA)**2 + 2*S*UGA(xA)**2 - 2*S*UGA(xA): LG})\n",
    "       .subs({-2.0*M*UGA(xA) - 1.0*M + 1.0*P*UGA(xA)**2 - 2.0*P*UGA(xA) + 1.0*P + 3.0*R*UGA(xA)**2 - 3.0*S*UGA(xA)**2 + 3.0*S*UGA(xA) - 1.0*T*UGA(xA)**2 + 1.0*T*UGA(xA): LG + 0.5 * FG})\n",
    "       .subs({M*UGA(xA) - M + P*UGA(xA)**2 - 2*P*UGA(xA) + P*UgA(xA)**2 - 2*P*UgA(xA) + 2*P - T*UGA(xA)**2 + T*UGA(xA) - T*UgA(xA)**2 + T*UgA(xA): 0.5 * FG - 0.5 * LG + 0.5 * Fg - 0.5 * Lg}))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 186,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2.0 P \\operatorname{UgA}^{2}{\\left(xA \\right)} - 4.0 P \\operatorname{UgA}{\\left(xA \\right)} + 2.0 P + 2.0 R \\operatorname{UgA}^{2}{\\left(xA \\right)} - 2.0 S \\operatorname{UgA}^{2}{\\left(xA \\right)} + 2.0 S \\operatorname{UgA}{\\left(xA \\right)} - 2.0 T \\operatorname{UgA}^{2}{\\left(xA \\right)} + 2.0 T \\operatorname{UgA}{\\left(xA \\right)} + x_{1} \\left(- 2.0 M + 2.0 P \\operatorname{UGA}^{2}{\\left(xA \\right)} - 4.0 P \\operatorname{UGA}{\\left(xA \\right)} - 2.0 P \\operatorname{UgA}^{2}{\\left(xA \\right)} + 4.0 P \\operatorname{UgA}{\\left(xA \\right)} + 2.0 R \\operatorname{UGA}^{2}{\\left(xA \\right)} - 2.0 R \\operatorname{UgA}^{2}{\\left(xA \\right)} - 2.0 S \\operatorname{UGA}^{2}{\\left(xA \\right)} + 2.0 S \\operatorname{UGA}{\\left(xA \\right)} + 2.0 S \\operatorname{UgA}^{2}{\\left(xA \\right)} - 2.0 S \\operatorname{UgA}{\\left(xA \\right)} - 2.0 T \\operatorname{UGA}^{2}{\\left(xA \\right)} + 2.0 T \\operatorname{UGA}{\\left(xA \\right)} + 2.0 T \\operatorname{UgA}^{2}{\\left(xA \\right)} - 2.0 T \\operatorname{UgA}{\\left(xA \\right)}\\right) + x_{2} \\left(- 2.0 M + 2.0 P \\operatorname{UGA}^{2}{\\left(xA \\right)} - 4.0 P \\operatorname{UGA}{\\left(xA \\right)} - 2.0 P \\operatorname{UgA}^{2}{\\left(xA \\right)} + 4.0 P \\operatorname{UgA}{\\left(xA \\right)} + 2.0 R \\operatorname{UGA}^{2}{\\left(xA \\right)} - 2.0 R \\operatorname{UgA}^{2}{\\left(xA \\right)} - 2.0 S \\operatorname{UGA}^{2}{\\left(xA \\right)} + 2.0 S \\operatorname{UGA}{\\left(xA \\right)} + 2.0 S \\operatorname{UgA}^{2}{\\left(xA \\right)} - 2.0 S \\operatorname{UgA}{\\left(xA \\right)} - 2.0 T \\operatorname{UGA}^{2}{\\left(xA \\right)} + 2.0 T \\operatorname{UGA}{\\left(xA \\right)} + 2.0 T \\operatorname{UgA}^{2}{\\left(xA \\right)} - 2.0 T \\operatorname{UgA}{\\left(xA \\right)}\\right)$"
      ],
      "text/plain": [
       "2.0*P*UgA(xA)**2 - 4.0*P*UgA(xA) + 2.0*P + 2.0*R*UgA(xA)**2 - 2.0*S*UgA(xA)**2 + 2.0*S*UgA(xA) - 2.0*T*UgA(xA)**2 + 2.0*T*UgA(xA) + x1*(-2.0*M + 2.0*P*UGA(xA)**2 - 4.0*P*UGA(xA) - 2.0*P*UgA(xA)**2 + 4.0*P*UgA(xA) + 2.0*R*UGA(xA)**2 - 2.0*R*UgA(xA)**2 - 2.0*S*UGA(xA)**2 + 2.0*S*UGA(xA) + 2.0*S*UgA(xA)**2 - 2.0*S*UgA(xA) - 2.0*T*UGA(xA)**2 + 2.0*T*UGA(xA) + 2.0*T*UgA(xA)**2 - 2.0*T*UgA(xA)) + x2*(-2.0*M + 2.0*P*UGA(xA)**2 - 4.0*P*UGA(xA) - 2.0*P*UgA(xA)**2 + 4.0*P*UgA(xA) + 2.0*R*UGA(xA)**2 - 2.0*R*UgA(xA)**2 - 2.0*S*UGA(xA)**2 + 2.0*S*UGA(xA) + 2.0*S*UgA(xA)**2 - 2.0*S*UgA(xA) - 2.0*T*UGA(xA)**2 + 2.0*T*UGA(xA) + 2.0*T*UgA(xA)**2 - 2.0*T*UgA(xA))"
      ]
     },
     "execution_count": 186,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(sym.simplify(d).subs({x1 + x3: xA}), x1, x2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 363,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = sym.Matrix([f1, f2, f3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 364,
   "metadata": {},
   "outputs": [],
   "source": [
    "F_jac = F.jacobian([x1, x2, x3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 378,
   "metadata": {},
   "outputs": [],
   "source": [
    "_F_jac = F.subs({epsilon: 0, M: 0}).jacobian([x1, x2, x3])\n",
    "_e1, _e2, _e3 = (_F_jac.subs({x2: 1 - x1, x3: 0})\n",
    "       .doit()\n",
    "       .subs({sym.Derivative(UGA(x1), x1): 0, UgA(x1): Ug})\n",
    "       .subs({UGA(x1): 1})\n",
    "       .subs({x1: 1})\n",
    "       .eigenvals()\n",
    "       .keys())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 379,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{1}{2}$"
      ],
      "text/plain": [
       "-1/2"
      ]
     },
     "execution_count": 379,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_e1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 386,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{3 P Ug^{2} - 6 P Ug + 3 P + 3 R Ug^{2} - 6 R - 3 S Ug^{2} + 3 S Ug - 3 T Ug^{2} + 3 T Ug - \\sqrt{P^{2} Ug^{4} - 4 P^{2} Ug^{3} + 6 P^{2} Ug^{2} - 4 P^{2} Ug + P^{2} + 2 P R Ug^{4} - 4 P R Ug^{3} - 2 P R Ug^{2} + 8 P R Ug - 4 P R - 2 P S Ug^{4} + 6 P S Ug^{3} - 6 P S Ug^{2} + 2 P S Ug - 2 P T Ug^{4} + 6 P T Ug^{3} - 6 P T Ug^{2} + 2 P T Ug + R^{2} Ug^{4} + 4 R^{2} Ug^{2} + 4 R^{2} - 2 R S Ug^{4} + 2 R S Ug^{3} - 4 R S Ug^{2} + 4 R S Ug - 2 R T Ug^{4} + 2 R T Ug^{3} + 4 R T Ug^{2} - 4 R T Ug + S^{2} Ug^{4} - 2 S^{2} Ug^{3} + S^{2} Ug^{2} + 2 S T Ug^{4} - 4 S T Ug^{3} + 2 S T Ug^{2} + T^{2} Ug^{4} - 2 T^{2} Ug^{3} + T^{2} Ug^{2}}}{8 R}$"
      ],
      "text/plain": [
       "(3*P*Ug**2 - 6*P*Ug + 3*P + 3*R*Ug**2 - 6*R - 3*S*Ug**2 + 3*S*Ug - 3*T*Ug**2 + 3*T*Ug - sqrt(P**2*Ug**4 - 4*P**2*Ug**3 + 6*P**2*Ug**2 - 4*P**2*Ug + P**2 + 2*P*R*Ug**4 - 4*P*R*Ug**3 - 2*P*R*Ug**2 + 8*P*R*Ug - 4*P*R - 2*P*S*Ug**4 + 6*P*S*Ug**3 - 6*P*S*Ug**2 + 2*P*S*Ug - 2*P*T*Ug**4 + 6*P*T*Ug**3 - 6*P*T*Ug**2 + 2*P*T*Ug + R**2*Ug**4 + 4*R**2*Ug**2 + 4*R**2 - 2*R*S*Ug**4 + 2*R*S*Ug**3 - 4*R*S*Ug**2 + 4*R*S*Ug - 2*R*T*Ug**4 + 2*R*T*Ug**3 + 4*R*T*Ug**2 - 4*R*T*Ug + S**2*Ug**4 - 2*S**2*Ug**3 + S**2*Ug**2 + 2*S*T*Ug**4 - 4*S*T*Ug**3 + 2*S*T*Ug**2 + T**2*Ug**4 - 2*T**2*Ug**3 + T**2*Ug**2))/(8*R)"
      ]
     },
     "execution_count": 386,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_e2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 382,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle P^{2} Ug^{4} - 4 P^{2} Ug^{3} + 6 P^{2} Ug^{2} - 4 P^{2} Ug + P^{2} + 2 P R Ug^{4} - 4 P R Ug^{3} - 2 P R Ug^{2} + 8 P R Ug - 4 P R - 2 P S Ug^{4} + 6 P S Ug^{3} - 6 P S Ug^{2} + 2 P S Ug - 2 P T Ug^{4} + 6 P T Ug^{3} - 6 P T Ug^{2} + 2 P T Ug + R^{2} Ug^{4} + 4 R^{2} Ug^{2} + 4 R^{2} - 2 R S Ug^{4} + 2 R S Ug^{3} - 4 R S Ug^{2} + 4 R S Ug - 2 R T Ug^{4} + 2 R T Ug^{3} + 4 R T Ug^{2} - 4 R T Ug + S^{2} Ug^{4} - 2 S^{2} Ug^{3} + S^{2} Ug^{2} + 2 S T Ug^{4} - 4 S T Ug^{3} + 2 S T Ug^{2} + T^{2} Ug^{4} - 2 T^{2} Ug^{3} + T^{2} Ug^{2}$"
      ],
      "text/plain": [
       "P**2*Ug**4 - 4*P**2*Ug**3 + 6*P**2*Ug**2 - 4*P**2*Ug + P**2 + 2*P*R*Ug**4 - 4*P*R*Ug**3 - 2*P*R*Ug**2 + 8*P*R*Ug - 4*P*R - 2*P*S*Ug**4 + 6*P*S*Ug**3 - 6*P*S*Ug**2 + 2*P*S*Ug - 2*P*T*Ug**4 + 6*P*T*Ug**3 - 6*P*T*Ug**2 + 2*P*T*Ug + R**2*Ug**4 + 4*R**2*Ug**2 + 4*R**2 - 2*R*S*Ug**4 + 2*R*S*Ug**3 - 4*R*S*Ug**2 + 4*R*S*Ug - 2*R*T*Ug**4 + 2*R*T*Ug**3 + 4*R*T*Ug**2 - 4*R*T*Ug + S**2*Ug**4 - 2*S**2*Ug**3 + S**2*Ug**2 + 2*S*T*Ug**4 - 4*S*T*Ug**3 + 2*S*T*Ug**2 + T**2*Ug**4 - 2*T**2*Ug**3 + T**2*Ug**2"
      ]
     },
     "execution_count": 382,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(P**2*Ug**4 - 4*P**2*Ug**3 + 6*P**2*Ug**2 - 4*P**2*Ug + P**2 + 2*P*R*Ug**4 - 4*P*R*Ug**3 - 2*P*R*Ug**2 + 8*P*R*Ug - 4*P*R - 2*P*S*Ug**4 + 6*P*S*Ug**3 - 6*P*S*Ug**2 + 2*P*S*Ug - 2*P*T*Ug**4 + 6*P*T*Ug**3 - 6*P*T*Ug**2 + 2*P*T*Ug + R**2*Ug**4 + 4*R**2*Ug**2 + 4*R**2 - 2*R*S*Ug**4 + 2*R*S*Ug**3 - 4*R*S*Ug**2 + 4*R*S*Ug - 2*R*T*Ug**4 + 2*R*T*Ug**3 + 4*R*T*Ug**2 - 4*R*T*Ug + S**2*Ug**4 - 2*S**2*Ug**3 + S**2*Ug**2 + 2*S*T*Ug**4 - 4*S*T*Ug**3 + 2*S*T*Ug**2 + T**2*Ug**4 - 2*T**2*Ug**3 + T**2*Ug**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 366,
   "metadata": {},
   "outputs": [],
   "source": [
    "FG = sym.symbols(\"FG\", real=True, positive=True)\n",
    "Fg = sym.symbols(\"Fg\", real=True, positive=True)\n",
    "\n",
    "Ug = sym.symbols(\"Ug\", real=True, nonnegative=True)\n",
    "\n",
    "\n",
    "simplified_F_jac = sym.zeros(3, 3)\n",
    "for i in range(3):\n",
    "    for j in range(3):\n",
    "        simplified_F_jac[i,j] = (sym.factor(sym.nsimplify(sym.cancel(F_jac[i,j].subs({x2: 1 - x1, x3: 0})\n",
    "           .doit()\n",
    "           .subs({sym.Derivative(UGA(x1), x1): 0, UgA(x1): Ug})\n",
    "           .subs({UGA(x1): 1})\n",
    "           .subs({x1: x1_star})\n",
    "           .subs({-2*M + 2*R: FG, -1.0 * M + 1.0 * R: 0.5 * FG}))), epsilon)\n",
    "           .subs({-4*P*Ug**2 + 8*P*Ug - 4*P - 3*R*Ug**2 + 3*S*Ug**2 - 3*S*Ug + 4*T*Ug**2 - 4*T*Ug: -2 * Fg + 0.5 * Lg})\n",
    "           .subs({-2*P*Ug**2 + 4*P*Ug - 2*P - R*Ug**2 + S*Ug**2 - S*Ug + 2*T*Ug**2 - 2*T*Ug: -Fg + 0.5 * Lg})\n",
    "           .subs({R*Ug**2 - S*Ug**2 + S*Ug: 0.5 * Lg})\n",
    "           .subs({2*P*Ug**2 - 4*P*Ug + 2*P + 2*R*Ug**2 - 2*S*Ug**2 + 2*S*Ug - 2*T*Ug**2 + 2*T*Ug: Fg})\n",
    "           .subs({4*P*Ug**2 - 8*P*Ug + 4*P - 4*T*Ug**2 + 4*T*Ug: 2*Fg - 2 * Lg})\n",
    "           .subs({-4*M + 4*R: 2*FG})\n",
    "           .subs({8*M - 8*R: -4*FG})\n",
    "           .subs({P*Ug**2 - 2*P*Ug + P - T*Ug**2 + T*Ug: 0.5 * Fg - 0.5 * Lg})\n",
    "           .subs({2*P*Ug**2 - 4*P*Ug + 2*P - 2*T*Ug**2 + 2*T*Ug: Fg - Lg})\n",
    "           .subs({6*P*Ug**2 - 12*P*Ug + 6*P + 6*R*Ug**2 - 6*S*Ug**2 + 6*S*Ug - 6*T*Ug**2 + 6*T*Ug: 3 * Fg}))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 367,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left[\\begin{matrix}- \\frac{2 FG - 2 Fg + 0.5 Lg + \\epsilon^{2} \\left(4 FG + 2 Fg - 2 Lg\\right) + \\epsilon \\left(2 FG + Fg\\right)}{2 FG \\left(2 \\epsilon + 1\\right)} & - \\frac{FG - 2 Fg + 0.5 Lg + \\epsilon^{2} \\left(2 Fg - 2 Lg\\right) + \\epsilon \\left(- 2 FG + Fg\\right)}{2 FG \\left(2 \\epsilon + 1\\right)} & - \\frac{FG + Fg \\epsilon - 0.5 Fg}{2 FG}\\\\\\frac{- Fg + 0.5 Lg + \\epsilon^{2} \\left(4 FG + 2 Fg - 2 Lg\\right) + \\epsilon \\left(- 2 FG + 3 Fg\\right)}{2 FG \\left(2 \\epsilon + 1\\right)} & \\frac{- FG - Fg + 0.5 Lg + \\epsilon^{2} \\left(2 Fg - 2 Lg\\right) + \\epsilon \\left(- 6 FG + 3 Fg\\right)}{2 FG \\left(2 \\epsilon + 1\\right)} & - \\frac{0.5 Fg + \\epsilon \\left(2 FG - Fg\\right)}{2 FG}\\\\- \\frac{0.5 Lg + \\epsilon \\left(Fg - Lg\\right)}{2 FG} & - \\frac{0.5 Lg + \\epsilon \\left(Fg - Lg\\right)}{2 FG} & - \\frac{FG + Fg \\epsilon - 0.5 Fg}{2 FG}\\end{matrix}\\right]$"
      ],
      "text/plain": [
       "Matrix([\n",
       "[-(2*FG - 2*Fg + 0.5*Lg + epsilon**2*(4*FG + 2*Fg - 2*Lg) + epsilon*(2*FG + Fg))/(2*FG*(2*epsilon + 1)), -(FG - 2*Fg + 0.5*Lg + epsilon**2*(2*Fg - 2*Lg) + epsilon*(-2*FG + Fg))/(2*FG*(2*epsilon + 1)),     -(FG + Fg*epsilon - 0.5*Fg)/(2*FG)],\n",
       "[      (-Fg + 0.5*Lg + epsilon**2*(4*FG + 2*Fg - 2*Lg) + epsilon*(-2*FG + 3*Fg))/(2*FG*(2*epsilon + 1)), (-FG - Fg + 0.5*Lg + epsilon**2*(2*Fg - 2*Lg) + epsilon*(-6*FG + 3*Fg))/(2*FG*(2*epsilon + 1)), -(0.5*Fg + epsilon*(2*FG - Fg))/(2*FG)],\n",
       "[                                                                  -(0.5*Lg + epsilon*(Fg - Lg))/(2*FG),                                                           -(0.5*Lg + epsilon*(Fg - Lg))/(2*FG),     -(FG + Fg*epsilon - 0.5*Fg)/(2*FG)]])"
      ]
     },
     "execution_count": 367,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplified_F_jac"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 333,
   "metadata": {},
   "outputs": [],
   "source": [
    "eigenvals = simplified_F_jac.eigenvals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 337,
   "metadata": {},
   "outputs": [],
   "source": [
    "e1, e2, e3 = eigenvals.keys()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 338,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\epsilon - \\frac{1}{2}$"
      ],
      "text/plain": [
       "-epsilon - 1/2"
      ]
     },
     "execution_count": 338,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 351,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{6 FG + 2 Fg \\epsilon - 3 Fg + \\sqrt{4 FG^{2} + 32 FG Fg \\epsilon^{2} + 8 FG Fg \\epsilon - 4 FG Fg - 32 FG Lg \\epsilon^{2} + 8 FG Lg + 4 Fg^{2} \\epsilon^{2} + 4 Fg^{2} \\epsilon + Fg^{2}}}{8 FG}$"
      ],
      "text/plain": [
       "-(6*FG + 2*Fg*epsilon - 3*Fg + sqrt(4*FG**2 + 32*FG*Fg*epsilon**2 + 8*FG*Fg*epsilon - 4*FG*Fg - 32*FG*Lg*epsilon**2 + 8*FG*Lg + 4*Fg**2*epsilon**2 + 4*Fg**2*epsilon + Fg**2))/(8*FG)"
      ]
     },
     "execution_count": 351,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 344,
   "metadata": {},
   "outputs": [],
   "source": [
    "r = 6*FG + 2*Fg*epsilon - 3*Fg\n",
    "delta = 4*FG**2 + 32*FG*Fg*epsilon**2 + 8*FG*Fg*epsilon - 4*FG*Fg - 32*FG*Lg*epsilon**2 + 8*FG*Lg + 4*Fg**2*epsilon**2 + 4*Fg**2*epsilon + Fg**2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 376,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 8 \\left(4 FG^{2} + FG \\left(- 4 Fg \\epsilon^{2} + 2 Fg \\epsilon - 4 Fg + 4 Lg \\epsilon^{2} - Lg\\right) - 2 Fg^{2} \\epsilon + Fg^{2}\\right)$"
      ],
      "text/plain": [
       "-8*(4*FG**2 + FG*(-4*Fg*epsilon**2 + 2*Fg*epsilon - 4*Fg + 4*Lg*epsilon**2 - Lg) - 2*Fg**2*epsilon + Fg**2)"
      ]
     },
     "execution_count": 376,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(delta - r**2, FG)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 340,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{6 FG + 2 Fg \\epsilon - 3 Fg - \\sqrt{4 FG^{2} + 32 FG Fg \\epsilon^{2} + 8 FG Fg \\epsilon - 4 FG Fg - 32 FG Lg \\epsilon^{2} + 8 FG Lg + 4 Fg^{2} \\epsilon^{2} + 4 Fg^{2} \\epsilon + Fg^{2}}}{8 FG}$"
      ],
      "text/plain": [
       "-(6*FG + 2*Fg*epsilon - 3*Fg - sqrt(4*FG**2 + 32*FG*Fg*epsilon**2 + 8*FG*Fg*epsilon - 4*FG*Fg - 32*FG*Lg*epsilon**2 + 8*FG*Lg + 4*Fg**2*epsilon**2 + 4*Fg**2*epsilon + Fg**2))/(8*FG)"
      ]
     },
     "execution_count": 340,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 4.0 \\left(0.5 P + Ug^{2} \\left(0.5 P + 0.5 R - 0.5 S - 0.5 T\\right) + Ug \\left(- 1.0 P + 0.5 S + 0.5 T\\right)\\right)$"
      ],
      "text/plain": [
       "4.0*(0.5*P + Ug**2*(0.5*P + 0.5*R - 0.5*S - 0.5*T) + Ug*(-1.0*P + 0.5*S + 0.5*T))"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(sym.expand(1.0*epsilon*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) + \\\n",
    "           2*(0.5 - 0.5*epsilon)*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) + \\\n",
    "           1.0*(2*R*Ug + 2*S*(1 - Ug))*Ug), Ug)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 85,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 1.0 P Ug^{2} \\epsilon + 2.0 P Ug \\epsilon - 1.0 P \\epsilon - 0.5 R Ug^{2} + 0.5 S Ug^{2} - 0.5 S Ug + 1.0 T Ug^{2} \\epsilon - 1.0 T Ug \\epsilon$"
      ],
      "text/plain": [
       "-1.0*P*Ug**2*epsilon + 2.0*P*Ug*epsilon - 1.0*P*epsilon - 0.5*R*Ug**2 + 0.5*S*Ug**2 - 0.5*S*Ug + 1.0*T*Ug**2*epsilon - 1.0*T*Ug*epsilon"
      ]
     },
     "execution_count": 85,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.expand(-0.25*Ug*(2*R*Ug + 2*S*(1 - Ug)) - \\\n",
    "           0.5*epsilon*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 71,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\left(\\frac{FG \\left(1 - \\epsilon\\right)}{2.0 \\epsilon + 1.0} + 0.5 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)\\right) \\left(- FG + 1.0 Ug \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) + 1.0 \\epsilon \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right) + 2 \\left(0.5 - 0.5 \\epsilon\\right) \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right)\\right)}{\\left(\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + \\frac{FG \\left(1 - \\epsilon\\right)}{2.0 \\epsilon + 1.0} + 1.0 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)\\right)^{2}} + \\frac{0.5 FG - 0.25 Ug \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) - 0.5 \\epsilon \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right)}{\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + \\frac{FG \\left(1 - \\epsilon\\right)}{2.0 \\epsilon + 1.0} + 1.0 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)}$"
      ],
      "text/plain": [
       "(FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 0.5*FG*(1 - 1/(2.0*epsilon + 1.0)))*(-FG + 1.0*Ug*(2*R*Ug + 2*S*(1 - Ug)) + 1.0*epsilon*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) + 2*(0.5 - 0.5*epsilon)*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug))/(FG*epsilon/(2.0*epsilon + 1.0) + FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 1.0*FG*(1 - 1/(2.0*epsilon + 1.0)))**2 + (0.5*FG - 0.25*Ug*(2*R*Ug + 2*S*(1 - Ug)) - 0.5*epsilon*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug))/(FG*epsilon/(2.0*epsilon + 1.0) + FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 1.0*FG*(1 - 1/(2.0*epsilon + 1.0)))"
      ]
     },
     "execution_count": 71,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(F_jac[0,1].subs({x2: 1 - x1, x3: 0})\n",
    "           .doit()\n",
    "           .subs({sym.Derivative(UGA(x1), x1): 0, UgA(x1): Ug})\n",
    "           .subs({UGA(x1): 1})\n",
    "           .subs({x1: x1_star})\n",
    "           .subs({-2*M + 2*R: FG, -1.0 * M + 1.0 * R: 0.5 * FG})\n",
    "           .subs({2.0*M - 2.0*R: -FG}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\left(\\frac{FG \\left(1 - \\epsilon\\right)}{2.0 \\epsilon + 1.0} + 0.5 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)\\right) \\left(- FG \\left(0.5 - 0.5 \\epsilon\\right) - 1.0 Ug \\epsilon \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) - 2 Ug \\left(0.5 - 0.5 \\epsilon\\right) \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) + 1.0 Ug \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) + 1.0 \\epsilon \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right) + 2 \\left(0.5 - 0.5 \\epsilon\\right) \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right) - 1.0 \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right) - \\frac{FG \\left(0.5 - 0.5 \\epsilon\\right) \\left(2.0 \\epsilon + 1.0\\right) - FG \\left(1 - \\epsilon\\right) \\left(2.0 \\epsilon + 1.0\\right)}{2.0 \\epsilon + 1.0}\\right)}{\\left(\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + \\frac{FG \\left(1 - \\epsilon\\right)}{2.0 \\epsilon + 1.0} + 1.0 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)\\right)^{2}} + \\frac{- 0.25 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right) \\left(2.0 \\epsilon + 1.0\\right) + Ug \\left(0.5 - 0.5 \\epsilon\\right) \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) - 0.25 Ug \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) - 0.5 \\epsilon \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right) + 0.25 \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right) + \\frac{FG \\left(0.5 - 0.5 \\epsilon\\right) \\left(2.0 \\epsilon + 1.0\\right) - FG \\left(1 - \\epsilon\\right) \\left(2.0 \\epsilon + 1.0\\right)}{2.0 \\epsilon + 1.0}}{\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + \\frac{FG \\left(1 - \\epsilon\\right)}{2.0 \\epsilon + 1.0} + 1.0 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)}$"
      ],
      "text/plain": [
       "(FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 0.5*FG*(1 - 1/(2.0*epsilon + 1.0)))*(-FG*(0.5 - 0.5*epsilon) - 1.0*Ug*epsilon*(2*R*Ug + 2*S*(1 - Ug)) - 2*Ug*(0.5 - 0.5*epsilon)*(2*R*Ug + 2*S*(1 - Ug)) + 1.0*Ug*(2*R*Ug + 2*S*(1 - Ug)) + 1.0*epsilon*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) + 2*(0.5 - 0.5*epsilon)*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) - 1.0*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) - (FG*(0.5 - 0.5*epsilon)*(2.0*epsilon + 1.0) - FG*(1 - epsilon)*(2.0*epsilon + 1.0))/(2.0*epsilon + 1.0))/(FG*epsilon/(2.0*epsilon + 1.0) + FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 1.0*FG*(1 - 1/(2.0*epsilon + 1.0)))**2 + (-0.25*FG*(1 - 1/(2.0*epsilon + 1.0))*(2.0*epsilon + 1.0) + Ug*(0.5 - 0.5*epsilon)*(2*R*Ug + 2*S*(1 - Ug)) - 0.25*Ug*(2*R*Ug + 2*S*(1 - Ug)) - 0.5*epsilon*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) + 0.25*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) + (FG*(0.5 - 0.5*epsilon)*(2.0*epsilon + 1.0) - FG*(1 - epsilon)*(2.0*epsilon + 1.0))/(2.0*epsilon + 1.0))/(FG*epsilon/(2.0*epsilon + 1.0) + FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 1.0*FG*(1 - 1/(2.0*epsilon + 1.0)))"
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(F_jac[0,2].subs({x2: 1 - x1, x3: 0})\n",
    "           .doit()\n",
    "           .subs({sym.Derivative(UGA(x1), x1): 0, UgA(x1): Ug})\n",
    "           .subs({UGA(x1): 1})\n",
    "           .subs({x1: x1_star})\n",
    "           .subs({-2*M + 2*R: FG, -1.0 * M + 1.0 * R: 0.5 * FG}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\left(\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + 0.5 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)\\right) \\left(- FG \\epsilon - FG \\left(1 - \\epsilon\\right) + 1.0 Ug \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) + 1.0 \\epsilon \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right) + 2 \\left(0.5 - 0.5 \\epsilon\\right) \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right)\\right)}{\\left(\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + \\frac{FG \\left(1 - \\epsilon\\right)}{2.0 \\epsilon + 1.0} + 1.0 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)\\right)^{2}} + \\frac{FG \\epsilon - 0.25 Ug \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) - \\left(0.5 - 0.5 \\epsilon\\right) \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right)}{\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + \\frac{FG \\left(1 - \\epsilon\\right)}{2.0 \\epsilon + 1.0} + 1.0 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)}$"
      ],
      "text/plain": [
       "(FG*epsilon/(2.0*epsilon + 1.0) + 0.5*FG*(1 - 1/(2.0*epsilon + 1.0)))*(-FG*epsilon - FG*(1 - epsilon) + 1.0*Ug*(2*R*Ug + 2*S*(1 - Ug)) + 1.0*epsilon*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) + 2*(0.5 - 0.5*epsilon)*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug))/(FG*epsilon/(2.0*epsilon + 1.0) + FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 1.0*FG*(1 - 1/(2.0*epsilon + 1.0)))**2 + (FG*epsilon - 0.25*Ug*(2*R*Ug + 2*S*(1 - Ug)) - (0.5 - 0.5*epsilon)*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug))/(FG*epsilon/(2.0*epsilon + 1.0) + FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 1.0*FG*(1 - 1/(2.0*epsilon + 1.0)))"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(F_jac[1,0].subs({x2: 1 - x1, x3: 0})\n",
    "           .doit()\n",
    "           .subs({sym.Derivative(UGA(x1), x1): 0, UgA(x1): Ug})\n",
    "           .subs({UGA(x1): 1})\n",
    "           .subs({x1: x1_star})\n",
    "           .subs({-2*M + 2*R: FG, -1.0 * M + 1.0 * R: 0.5 * FG}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\left(\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + 0.5 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)\\right) \\left(2.0 M - 2.0 R + 1.0 Ug \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) + 1.0 \\epsilon \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right) + 2 \\left(0.5 - 0.5 \\epsilon\\right) \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right)\\right)}{\\left(\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + \\frac{FG \\left(1 - \\epsilon\\right)}{2.0 \\epsilon + 1.0} + 1.0 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)\\right)^{2}} + \\frac{0.5 FG - 0.25 Ug \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) - \\left(0.5 - 0.5 \\epsilon\\right) \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right)}{\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + \\frac{FG \\left(1 - \\epsilon\\right)}{2.0 \\epsilon + 1.0} + 1.0 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)} - 1$"
      ],
      "text/plain": [
       "(FG*epsilon/(2.0*epsilon + 1.0) + 0.5*FG*(1 - 1/(2.0*epsilon + 1.0)))*(2.0*M - 2.0*R + 1.0*Ug*(2*R*Ug + 2*S*(1 - Ug)) + 1.0*epsilon*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) + 2*(0.5 - 0.5*epsilon)*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug))/(FG*epsilon/(2.0*epsilon + 1.0) + FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 1.0*FG*(1 - 1/(2.0*epsilon + 1.0)))**2 + (0.5*FG - 0.25*Ug*(2*R*Ug + 2*S*(1 - Ug)) - (0.5 - 0.5*epsilon)*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug))/(FG*epsilon/(2.0*epsilon + 1.0) + FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 1.0*FG*(1 - 1/(2.0*epsilon + 1.0))) - 1"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(F_jac[1,1].subs({x2: 1 - x1, x3: 0})\n",
    "           .doit()\n",
    "           .subs({sym.Derivative(UGA(x1), x1): 0, UgA(x1): Ug})\n",
    "           .subs({UGA(x1): 1})\n",
    "           .subs({x1: x1_star})\n",
    "           .subs({-2*M + 2*R: FG, -1.0 * M + 1.0 * R: 0.5 * FG}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{\\left(\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + 0.5 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)\\right) \\left(- FG \\left(0.5 - 0.5 \\epsilon\\right) - 1.0 Ug \\epsilon \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) - 2 Ug \\left(0.5 - 0.5 \\epsilon\\right) \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) + 1.0 Ug \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) + 1.0 \\epsilon \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right) + 2 \\left(0.5 - 0.5 \\epsilon\\right) \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right) - 1.0 \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right) - \\frac{FG \\left(0.5 - 0.5 \\epsilon\\right) \\left(2.0 \\epsilon + 1.0\\right) - FG \\left(1 - \\epsilon\\right) \\left(2.0 \\epsilon + 1.0\\right)}{2.0 \\epsilon + 1.0}\\right)}{\\left(\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + \\frac{FG \\left(1 - \\epsilon\\right)}{2.0 \\epsilon + 1.0} + 1.0 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)\\right)^{2}} + \\frac{- 0.5 FG \\epsilon - 0.25 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right) \\left(2.0 \\epsilon + 1.0\\right) + 0.5 Ug \\epsilon \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) - 0.25 Ug \\left(2 R Ug + 2 S \\left(1 - Ug\\right)\\right) - \\left(0.5 - 0.5 \\epsilon\\right) \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right) + 0.25 \\left(1 - Ug\\right) \\left(2 P \\left(1 - Ug\\right) + 2 T Ug\\right)}{\\frac{FG \\epsilon}{2.0 \\epsilon + 1.0} + \\frac{FG \\left(1 - \\epsilon\\right)}{2.0 \\epsilon + 1.0} + 1.0 FG \\left(1 - \\frac{1}{2.0 \\epsilon + 1.0}\\right)}$"
      ],
      "text/plain": [
       "(FG*epsilon/(2.0*epsilon + 1.0) + 0.5*FG*(1 - 1/(2.0*epsilon + 1.0)))*(-FG*(0.5 - 0.5*epsilon) - 1.0*Ug*epsilon*(2*R*Ug + 2*S*(1 - Ug)) - 2*Ug*(0.5 - 0.5*epsilon)*(2*R*Ug + 2*S*(1 - Ug)) + 1.0*Ug*(2*R*Ug + 2*S*(1 - Ug)) + 1.0*epsilon*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) + 2*(0.5 - 0.5*epsilon)*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) - 1.0*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) - (FG*(0.5 - 0.5*epsilon)*(2.0*epsilon + 1.0) - FG*(1 - epsilon)*(2.0*epsilon + 1.0))/(2.0*epsilon + 1.0))/(FG*epsilon/(2.0*epsilon + 1.0) + FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 1.0*FG*(1 - 1/(2.0*epsilon + 1.0)))**2 + (-0.5*FG*epsilon - 0.25*FG*(1 - 1/(2.0*epsilon + 1.0))*(2.0*epsilon + 1.0) + 0.5*Ug*epsilon*(2*R*Ug + 2*S*(1 - Ug)) - 0.25*Ug*(2*R*Ug + 2*S*(1 - Ug)) - (0.5 - 0.5*epsilon)*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) + 0.25*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug))/(FG*epsilon/(2.0*epsilon + 1.0) + FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 1.0*FG*(1 - 1/(2.0*epsilon + 1.0)))"
      ]
     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(F_jac[1,2].subs({x2: 1 - x1, x3: 0})\n",
    "           .doit()\n",
    "           .subs({sym.Derivative(UGA(x1), x1): 0, UgA(x1): Ug})\n",
    "           .subs({UGA(x1): 1})\n",
    "           .subs({x1: x1_star})\n",
    "           .subs({-2*M + 2*R: FG, -1.0 * M + 1.0 * R: 0.5 * FG}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-4.0*(1.0*x1 + 1.0*x3)*(1.0*x1 + 1.0*x3 - 1.0)"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor((-4.0*x1**2 - 8.0*x1*x3 + 4.0*x1 - 4.0*x3**2 + 4.0*x3))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 173,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'(-FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 0.5*FG*(1 - 1/(2.0*epsilon + 1.0)))*(FG*epsilon + FG*(1 - epsilon) + 1.0*Ug*(2*R*Ug + 2*S*(1 - Ug)) + 1.0*epsilon*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug) + 2*(0.5 - 0.5*epsilon)*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug))/(-FG*epsilon/(2.0*epsilon + 1.0) - FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 1.0*FG*(1 - 1/(2.0*epsilon + 1.0)))**2 + (-FG*(1 - epsilon) - 0.25*Ug*(2*R*Ug + 2*S*(1 - Ug)) - 0.5*epsilon*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug))/(-FG*epsilon/(2.0*epsilon + 1.0) - FG*(1 - epsilon)/(2.0*epsilon + 1.0) + 1.0*FG*(1 - 1/(2.0*epsilon + 1.0))) - 1'"
      ]
     },
     "execution_count": 173,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "str(F_jac[0,0].subs({x2: 1 - x1, x3: 0})\n",
    "           .doit()\n",
    "           .subs({sym.Derivative(UGA(x1), x1): 0, UgA(x1): Ug})\n",
    "           .subs({UGA(x1): 1})\n",
    "           .subs({x1: x1_star})\n",
    "           .subs({-2.0*M + 2.0*R: FG, -1.0 * M + 1.0 * R: 0.5 * FG, 2*M - 2*R: FG})\n",
    "           .subs({-M + R: 0.5 * FG})\n",
    "           .subs({1.0*Ug*(2*R*Ug + 2*S*(1 - Ug)) + 1.0*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug): Fg}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 168,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 2.0 P Ug^{2} - 4.0 P Ug + 2.0 P + 2.0 R Ug^{2} - 2.0 S Ug^{2} + 2.0 S Ug - 2.0 T Ug^{2} + 2.0 T Ug$"
      ],
      "text/plain": [
       "2.0*P*Ug**2 - 4.0*P*Ug + 2.0*P + 2.0*R*Ug**2 - 2.0*S*Ug**2 + 2.0*S*Ug - 2.0*T*Ug**2 + 2.0*T*Ug"
      ]
     },
     "execution_count": 168,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.expand(1.0*Ug*(2*R*Ug + 2*S*(1 - Ug)) + 1.0*(1 - Ug)*(2*P*(1 - Ug) + 2*T*Ug))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 161,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 0.5 P Ug^{2} - 1.0 P Ug + 0.5 P + 0.5 R Ug^{2} - 0.5 S Ug^{2} + 0.5 S Ug - 0.5 T Ug^{2} + 0.5 T Ug$"
      ],
      "text/plain": [
       "0.5*P*Ug**2 - 1.0*P*Ug + 0.5*P + 0.5*R*Ug**2 - 0.5*S*Ug**2 + 0.5*S*Ug - 0.5*T*Ug**2 + 0.5*T*Ug"
      ]
     },
     "execution_count": 161,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.expand(0.25 * Ug * ( 2 * R * Ug + 2 * S * (1 - Ug)) + 0.25 * (1 - Ug) * (2 * P * (1 - Ug) + 2 * T * Ug))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [],
   "source": [
    "eigen_vals = (F_jac.subs({x2: 1 - x1, x3: 0})\n",
    "           .doit()\n",
    "           .subs({sym.Derivative(UGA(x1), x1): 0, UgA(x1): Ug})\n",
    "           .subs({UGA(x1): 1})\n",
    "           .subs({x1: x1_star})\n",
    "           .subs({-2*M + 2*R: FG, -1.0 * M + 1.0 * R: 0.5 * FG})\n",
    "           .subs({2.0*M - 2.0*R: -FG})\n",
    "           .eigenvals())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "e1, e2, e3 = eigen_vals.keys()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\epsilon - \\frac{1}{2}$"
      ],
      "text/plain": [
       "-epsilon - 1/2"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{3 FG + 2 P Ug^{2} \\epsilon - 3 P Ug^{2} - 4 P Ug \\epsilon + 6 P Ug + 2 P \\epsilon - 3 P + 2 R Ug^{2} \\epsilon - 3 R Ug^{2} - 2 S Ug^{2} \\epsilon + 3 S Ug^{2} + 2 S Ug \\epsilon - 3 S Ug - 2 T Ug^{2} \\epsilon + 3 T Ug^{2} + 2 T Ug \\epsilon - 3 T Ug + \\sqrt{FG^{2} + 16 FG P Ug^{2} \\epsilon^{2} + 4 FG P Ug^{2} \\epsilon - 2 FG P Ug^{2} - 32 FG P Ug \\epsilon^{2} - 8 FG P Ug \\epsilon + 4 FG P Ug + 16 FG P \\epsilon^{2} + 4 FG P \\epsilon - 2 FG P + 4 FG R Ug^{2} \\epsilon + 2 FG R Ug^{2} - 4 FG S Ug^{2} \\epsilon - 2 FG S Ug^{2} + 4 FG S Ug \\epsilon + 2 FG S Ug - 16 FG T Ug^{2} \\epsilon^{2} - 4 FG T Ug^{2} \\epsilon + 2 FG T Ug^{2} + 16 FG T Ug \\epsilon^{2} + 4 FG T Ug \\epsilon - 2 FG T Ug + 4 P^{2} Ug^{4} \\epsilon^{2} + 4 P^{2} Ug^{4} \\epsilon + P^{2} Ug^{4} - 16 P^{2} Ug^{3} \\epsilon^{2} - 16 P^{2} Ug^{3} \\epsilon - 4 P^{2} Ug^{3} + 24 P^{2} Ug^{2} \\epsilon^{2} + 24 P^{2} Ug^{2} \\epsilon + 6 P^{2} Ug^{2} - 16 P^{2} Ug \\epsilon^{2} - 16 P^{2} Ug \\epsilon - 4 P^{2} Ug + 4 P^{2} \\epsilon^{2} + 4 P^{2} \\epsilon + P^{2} + 8 P R Ug^{4} \\epsilon^{2} + 8 P R Ug^{4} \\epsilon + 2 P R Ug^{4} - 16 P R Ug^{3} \\epsilon^{2} - 16 P R Ug^{3} \\epsilon - 4 P R Ug^{3} + 8 P R Ug^{2} \\epsilon^{2} + 8 P R Ug^{2} \\epsilon + 2 P R Ug^{2} - 8 P S Ug^{4} \\epsilon^{2} - 8 P S Ug^{4} \\epsilon - 2 P S Ug^{4} + 24 P S Ug^{3} \\epsilon^{2} + 24 P S Ug^{3} \\epsilon + 6 P S Ug^{3} - 24 P S Ug^{2} \\epsilon^{2} - 24 P S Ug^{2} \\epsilon - 6 P S Ug^{2} + 8 P S Ug \\epsilon^{2} + 8 P S Ug \\epsilon + 2 P S Ug - 8 P T Ug^{4} \\epsilon^{2} - 8 P T Ug^{4} \\epsilon - 2 P T Ug^{4} + 24 P T Ug^{3} \\epsilon^{2} + 24 P T Ug^{3} \\epsilon + 6 P T Ug^{3} - 24 P T Ug^{2} \\epsilon^{2} - 24 P T Ug^{2} \\epsilon - 6 P T Ug^{2} + 8 P T Ug \\epsilon^{2} + 8 P T Ug \\epsilon + 2 P T Ug + 4 R^{2} Ug^{4} \\epsilon^{2} + 4 R^{2} Ug^{4} \\epsilon + R^{2} Ug^{4} - 8 R S Ug^{4} \\epsilon^{2} - 8 R S Ug^{4} \\epsilon - 2 R S Ug^{4} + 8 R S Ug^{3} \\epsilon^{2} + 8 R S Ug^{3} \\epsilon + 2 R S Ug^{3} - 8 R T Ug^{4} \\epsilon^{2} - 8 R T Ug^{4} \\epsilon - 2 R T Ug^{4} + 8 R T Ug^{3} \\epsilon^{2} + 8 R T Ug^{3} \\epsilon + 2 R T Ug^{3} + 4 S^{2} Ug^{4} \\epsilon^{2} + 4 S^{2} Ug^{4} \\epsilon + S^{2} Ug^{4} - 8 S^{2} Ug^{3} \\epsilon^{2} - 8 S^{2} Ug^{3} \\epsilon - 2 S^{2} Ug^{3} + 4 S^{2} Ug^{2} \\epsilon^{2} + 4 S^{2} Ug^{2} \\epsilon + S^{2} Ug^{2} + 8 S T Ug^{4} \\epsilon^{2} + 8 S T Ug^{4} \\epsilon + 2 S T Ug^{4} - 16 S T Ug^{3} \\epsilon^{2} - 16 S T Ug^{3} \\epsilon - 4 S T Ug^{3} + 8 S T Ug^{2} \\epsilon^{2} + 8 S T Ug^{2} \\epsilon + 2 S T Ug^{2} + 4 T^{2} Ug^{4} \\epsilon^{2} + 4 T^{2} Ug^{4} \\epsilon + T^{2} Ug^{4} - 8 T^{2} Ug^{3} \\epsilon^{2} - 8 T^{2} Ug^{3} \\epsilon - 2 T^{2} Ug^{3} + 4 T^{2} Ug^{2} \\epsilon^{2} + 4 T^{2} Ug^{2} \\epsilon + T^{2} Ug^{2}}}{4 FG}$"
      ],
      "text/plain": [
       "-(3*FG + 2*P*Ug**2*epsilon - 3*P*Ug**2 - 4*P*Ug*epsilon + 6*P*Ug + 2*P*epsilon - 3*P + 2*R*Ug**2*epsilon - 3*R*Ug**2 - 2*S*Ug**2*epsilon + 3*S*Ug**2 + 2*S*Ug*epsilon - 3*S*Ug - 2*T*Ug**2*epsilon + 3*T*Ug**2 + 2*T*Ug*epsilon - 3*T*Ug + sqrt(FG**2 + 16*FG*P*Ug**2*epsilon**2 + 4*FG*P*Ug**2*epsilon - 2*FG*P*Ug**2 - 32*FG*P*Ug*epsilon**2 - 8*FG*P*Ug*epsilon + 4*FG*P*Ug + 16*FG*P*epsilon**2 + 4*FG*P*epsilon - 2*FG*P + 4*FG*R*Ug**2*epsilon + 2*FG*R*Ug**2 - 4*FG*S*Ug**2*epsilon - 2*FG*S*Ug**2 + 4*FG*S*Ug*epsilon + 2*FG*S*Ug - 16*FG*T*Ug**2*epsilon**2 - 4*FG*T*Ug**2*epsilon + 2*FG*T*Ug**2 + 16*FG*T*Ug*epsilon**2 + 4*FG*T*Ug*epsilon - 2*FG*T*Ug + 4*P**2*Ug**4*epsilon**2 + 4*P**2*Ug**4*epsilon + P**2*Ug**4 - 16*P**2*Ug**3*epsilon**2 - 16*P**2*Ug**3*epsilon - 4*P**2*Ug**3 + 24*P**2*Ug**2*epsilon**2 + 24*P**2*Ug**2*epsilon + 6*P**2*Ug**2 - 16*P**2*Ug*epsilon**2 - 16*P**2*Ug*epsilon - 4*P**2*Ug + 4*P**2*epsilon**2 + 4*P**2*epsilon + P**2 + 8*P*R*Ug**4*epsilon**2 + 8*P*R*Ug**4*epsilon + 2*P*R*Ug**4 - 16*P*R*Ug**3*epsilon**2 - 16*P*R*Ug**3*epsilon - 4*P*R*Ug**3 + 8*P*R*Ug**2*epsilon**2 + 8*P*R*Ug**2*epsilon + 2*P*R*Ug**2 - 8*P*S*Ug**4*epsilon**2 - 8*P*S*Ug**4*epsilon - 2*P*S*Ug**4 + 24*P*S*Ug**3*epsilon**2 + 24*P*S*Ug**3*epsilon + 6*P*S*Ug**3 - 24*P*S*Ug**2*epsilon**2 - 24*P*S*Ug**2*epsilon - 6*P*S*Ug**2 + 8*P*S*Ug*epsilon**2 + 8*P*S*Ug*epsilon + 2*P*S*Ug - 8*P*T*Ug**4*epsilon**2 - 8*P*T*Ug**4*epsilon - 2*P*T*Ug**4 + 24*P*T*Ug**3*epsilon**2 + 24*P*T*Ug**3*epsilon + 6*P*T*Ug**3 - 24*P*T*Ug**2*epsilon**2 - 24*P*T*Ug**2*epsilon - 6*P*T*Ug**2 + 8*P*T*Ug*epsilon**2 + 8*P*T*Ug*epsilon + 2*P*T*Ug + 4*R**2*Ug**4*epsilon**2 + 4*R**2*Ug**4*epsilon + R**2*Ug**4 - 8*R*S*Ug**4*epsilon**2 - 8*R*S*Ug**4*epsilon - 2*R*S*Ug**4 + 8*R*S*Ug**3*epsilon**2 + 8*R*S*Ug**3*epsilon + 2*R*S*Ug**3 - 8*R*T*Ug**4*epsilon**2 - 8*R*T*Ug**4*epsilon - 2*R*T*Ug**4 + 8*R*T*Ug**3*epsilon**2 + 8*R*T*Ug**3*epsilon + 2*R*T*Ug**3 + 4*S**2*Ug**4*epsilon**2 + 4*S**2*Ug**4*epsilon + S**2*Ug**4 - 8*S**2*Ug**3*epsilon**2 - 8*S**2*Ug**3*epsilon - 2*S**2*Ug**3 + 4*S**2*Ug**2*epsilon**2 + 4*S**2*Ug**2*epsilon + S**2*Ug**2 + 8*S*T*Ug**4*epsilon**2 + 8*S*T*Ug**4*epsilon + 2*S*T*Ug**4 - 16*S*T*Ug**3*epsilon**2 - 16*S*T*Ug**3*epsilon - 4*S*T*Ug**3 + 8*S*T*Ug**2*epsilon**2 + 8*S*T*Ug**2*epsilon + 2*S*T*Ug**2 + 4*T**2*Ug**4*epsilon**2 + 4*T**2*Ug**4*epsilon + T**2*Ug**4 - 8*T**2*Ug**3*epsilon**2 - 8*T**2*Ug**3*epsilon - 2*T**2*Ug**3 + 4*T**2*Ug**2*epsilon**2 + 4*T**2*Ug**2*epsilon + T**2*Ug**2))/(4*FG)"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 156,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{3 FG - 3 P Ug^{2} + 6 P Ug - 3 P - 3 R Ug^{2} + 3 S Ug^{2} - 3 S Ug + 3 T Ug^{2} - 3 T Ug + \\sqrt{FG^{2} - 2 FG P Ug^{2} + 4 FG P Ug - 2 FG P + 2 FG R Ug^{2} - 2 FG S Ug^{2} + 2 FG S Ug + 2 FG T Ug^{2} - 2 FG T Ug + P^{2} Ug^{4} - 4 P^{2} Ug^{3} + 6 P^{2} Ug^{2} - 4 P^{2} Ug + P^{2} + 2 P R Ug^{4} - 4 P R Ug^{3} + 2 P R Ug^{2} - 2 P S Ug^{4} + 6 P S Ug^{3} - 6 P S Ug^{2} + 2 P S Ug - 2 P T Ug^{4} + 6 P T Ug^{3} - 6 P T Ug^{2} + 2 P T Ug + R^{2} Ug^{4} - 2 R S Ug^{4} + 2 R S Ug^{3} - 2 R T Ug^{4} + 2 R T Ug^{3} + S^{2} Ug^{4} - 2 S^{2} Ug^{3} + S^{2} Ug^{2} + 2 S T Ug^{4} - 4 S T Ug^{3} + 2 S T Ug^{2} + T^{2} Ug^{4} - 2 T^{2} Ug^{3} + T^{2} Ug^{2}}}{4 FG}$"
      ],
      "text/plain": [
       "-(3*FG - 3*P*Ug**2 + 6*P*Ug - 3*P - 3*R*Ug**2 + 3*S*Ug**2 - 3*S*Ug + 3*T*Ug**2 - 3*T*Ug + sqrt(FG**2 - 2*FG*P*Ug**2 + 4*FG*P*Ug - 2*FG*P + 2*FG*R*Ug**2 - 2*FG*S*Ug**2 + 2*FG*S*Ug + 2*FG*T*Ug**2 - 2*FG*T*Ug + P**2*Ug**4 - 4*P**2*Ug**3 + 6*P**2*Ug**2 - 4*P**2*Ug + P**2 + 2*P*R*Ug**4 - 4*P*R*Ug**3 + 2*P*R*Ug**2 - 2*P*S*Ug**4 + 6*P*S*Ug**3 - 6*P*S*Ug**2 + 2*P*S*Ug - 2*P*T*Ug**4 + 6*P*T*Ug**3 - 6*P*T*Ug**2 + 2*P*T*Ug + R**2*Ug**4 - 2*R*S*Ug**4 + 2*R*S*Ug**3 - 2*R*T*Ug**4 + 2*R*T*Ug**3 + S**2*Ug**4 - 2*S**2*Ug**3 + S**2*Ug**2 + 2*S*T*Ug**4 - 4*S*T*Ug**3 + 2*S*T*Ug**2 + T**2*Ug**4 - 2*T**2*Ug**3 + T**2*Ug**2))/(4*FG)"
      ]
     },
     "execution_count": 156,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2.subs({epsilon: 0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 129,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'-(3*FG + 2*P*Ug**2*epsilon - 3*P*Ug**2 - 4*P*Ug*epsilon + 6*P*Ug + 2*P*epsilon - 3*P + 2*R*Ug**2*epsilon - 3*R*Ug**2 - 2*S*Ug**2*epsilon + 3*S*Ug**2 + 2*S*Ug*epsilon - 3*S*Ug - 2*T*Ug**2*epsilon + 3*T*Ug**2 + 2*T*Ug*epsilon - 3*T*Ug + sqrt(FG**2 + 16*FG*P*Ug**2*epsilon**2 + 4*FG*P*Ug**2*epsilon - 2*FG*P*Ug**2 - 32*FG*P*Ug*epsilon**2 - 8*FG*P*Ug*epsilon + 4*FG*P*Ug + 16*FG*P*epsilon**2 + 4*FG*P*epsilon - 2*FG*P + 4*FG*R*Ug**2*epsilon + 2*FG*R*Ug**2 - 4*FG*S*Ug**2*epsilon - 2*FG*S*Ug**2 + 4*FG*S*Ug*epsilon + 2*FG*S*Ug - 16*FG*T*Ug**2*epsilon**2 - 4*FG*T*Ug**2*epsilon + 2*FG*T*Ug**2 + 16*FG*T*Ug*epsilon**2 + 4*FG*T*Ug*epsilon - 2*FG*T*Ug + 4*P**2*Ug**4*epsilon**2 + 4*P**2*Ug**4*epsilon + P**2*Ug**4 - 16*P**2*Ug**3*epsilon**2 - 16*P**2*Ug**3*epsilon - 4*P**2*Ug**3 + 24*P**2*Ug**2*epsilon**2 + 24*P**2*Ug**2*epsilon + 6*P**2*Ug**2 - 16*P**2*Ug*epsilon**2 - 16*P**2*Ug*epsilon - 4*P**2*Ug + 4*P**2*epsilon**2 + 4*P**2*epsilon + P**2 + 8*P*R*Ug**4*epsilon**2 + 8*P*R*Ug**4*epsilon + 2*P*R*Ug**4 - 16*P*R*Ug**3*epsilon**2 - 16*P*R*Ug**3*epsilon - 4*P*R*Ug**3 + 8*P*R*Ug**2*epsilon**2 + 8*P*R*Ug**2*epsilon + 2*P*R*Ug**2 - 8*P*S*Ug**4*epsilon**2 - 8*P*S*Ug**4*epsilon - 2*P*S*Ug**4 + 24*P*S*Ug**3*epsilon**2 + 24*P*S*Ug**3*epsilon + 6*P*S*Ug**3 - 24*P*S*Ug**2*epsilon**2 - 24*P*S*Ug**2*epsilon - 6*P*S*Ug**2 + 8*P*S*Ug*epsilon**2 + 8*P*S*Ug*epsilon + 2*P*S*Ug - 8*P*T*Ug**4*epsilon**2 - 8*P*T*Ug**4*epsilon - 2*P*T*Ug**4 + 24*P*T*Ug**3*epsilon**2 + 24*P*T*Ug**3*epsilon + 6*P*T*Ug**3 - 24*P*T*Ug**2*epsilon**2 - 24*P*T*Ug**2*epsilon - 6*P*T*Ug**2 + 8*P*T*Ug*epsilon**2 + 8*P*T*Ug*epsilon + 2*P*T*Ug + 4*R**2*Ug**4*epsilon**2 + 4*R**2*Ug**4*epsilon + R**2*Ug**4 - 8*R*S*Ug**4*epsilon**2 - 8*R*S*Ug**4*epsilon - 2*R*S*Ug**4 + 8*R*S*Ug**3*epsilon**2 + 8*R*S*Ug**3*epsilon + 2*R*S*Ug**3 - 8*R*T*Ug**4*epsilon**2 - 8*R*T*Ug**4*epsilon - 2*R*T*Ug**4 + 8*R*T*Ug**3*epsilon**2 + 8*R*T*Ug**3*epsilon + 2*R*T*Ug**3 + 4*S**2*Ug**4*epsilon**2 + 4*S**2*Ug**4*epsilon + S**2*Ug**4 - 8*S**2*Ug**3*epsilon**2 - 8*S**2*Ug**3*epsilon - 2*S**2*Ug**3 + 4*S**2*Ug**2*epsilon**2 + 4*S**2*Ug**2*epsilon + S**2*Ug**2 + 8*S*T*Ug**4*epsilon**2 + 8*S*T*Ug**4*epsilon + 2*S*T*Ug**4 - 16*S*T*Ug**3*epsilon**2 - 16*S*T*Ug**3*epsilon - 4*S*T*Ug**3 + 8*S*T*Ug**2*epsilon**2 + 8*S*T*Ug**2*epsilon + 2*S*T*Ug**2 + 4*T**2*Ug**4*epsilon**2 + 4*T**2*Ug**4*epsilon + T**2*Ug**4 - 8*T**2*Ug**3*epsilon**2 - 8*T**2*Ug**3*epsilon - 2*T**2*Ug**3 + 4*T**2*Ug**2*epsilon**2 + 4*T**2*Ug**2*epsilon + T**2*Ug**2))/(4*FG)'"
      ]
     },
     "execution_count": 129,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "str(e2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {},
   "outputs": [],
   "source": [
    "r = (sym.factor(3*FG + 2*P*Ug**2*epsilon - 3*P*Ug**2 - 4*P*Ug*epsilon + 6*P*Ug + 2*P*epsilon - 3*P + 2*R*Ug**2*epsilon - 3*R*Ug**2 - 2*S*Ug**2*epsilon + 3*S*Ug**2 + 2*S*Ug*epsilon - 3*S*Ug - 2*T*Ug**2*epsilon + 3*T*Ug**2 + 2*T*Ug*epsilon - 3*T*Ug, epsilon, FG)\n",
    "    .subs({2*P*Ug**2 - 4*P*Ug + 2*P + 2*R*Ug**2 - 2*S*Ug**2 + 2*S*Ug - 2*T*Ug**2 + 2*T*Ug: Fg})\n",
    "    .subs({-3*P*Ug**2 + 6*P*Ug - 3*P - 3*R*Ug**2 + 3*S*Ug**2 - 3*S*Ug + 3*T*Ug**2 - 3*T*Ug: -1.5 * Fg + 6 * ((R - S) * Ug**2 + S * Ug)}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 3 \\left(P Ug^{2} - 2 P Ug + P + R Ug^{2} - S Ug^{2} + S Ug - T Ug^{2} + T Ug\\right)$"
      ],
      "text/plain": [
       "-3*(P*Ug**2 - 2*P*Ug + P + R*Ug**2 - S*Ug**2 + S*Ug - T*Ug**2 + T*Ug)"
      ]
     },
     "execution_count": 138,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(-3*P*Ug**2 + 6*P*Ug - 3*P - 3*R*Ug**2 + 3*S*Ug**2 - 3*S*Ug + 3*T*Ug**2 - 3*T*Ug)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 103,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{3 FG + 2 P Ug^{2} \\epsilon - 3 P Ug^{2} - 4 P Ug \\epsilon + 6 P Ug + 2 P \\epsilon - 3 P + 2 R Ug^{2} \\epsilon - 3 R Ug^{2} - 2 S Ug^{2} \\epsilon + 3 S Ug^{2} + 2 S Ug \\epsilon - 3 S Ug - 2 T Ug^{2} \\epsilon + 3 T Ug^{2} + 2 T Ug \\epsilon - 3 T Ug + \\sqrt{FG^{2} + 16 FG P Ug^{2} \\epsilon^{2} + 4 FG P Ug^{2} \\epsilon - 2 FG P Ug^{2} - 32 FG P Ug \\epsilon^{2} - 8 FG P Ug \\epsilon + 4 FG P Ug + 16 FG P \\epsilon^{2} + 4 FG P \\epsilon - 2 FG P + 4 FG R Ug^{2} \\epsilon + 2 FG R Ug^{2} - 4 FG S Ug^{2} \\epsilon - 2 FG S Ug^{2} + 4 FG S Ug \\epsilon + 2 FG S Ug - 16 FG T Ug^{2} \\epsilon^{2} - 4 FG T Ug^{2} \\epsilon + 2 FG T Ug^{2} + 16 FG T Ug \\epsilon^{2} + 4 FG T Ug \\epsilon - 2 FG T Ug + 4 P^{2} Ug^{4} \\epsilon^{2} + 4 P^{2} Ug^{4} \\epsilon + P^{2} Ug^{4} - 16 P^{2} Ug^{3} \\epsilon^{2} - 16 P^{2} Ug^{3} \\epsilon - 4 P^{2} Ug^{3} + 24 P^{2} Ug^{2} \\epsilon^{2} + 24 P^{2} Ug^{2} \\epsilon + 6 P^{2} Ug^{2} - 16 P^{2} Ug \\epsilon^{2} - 16 P^{2} Ug \\epsilon - 4 P^{2} Ug + 4 P^{2} \\epsilon^{2} + 4 P^{2} \\epsilon + P^{2} + 8 P R Ug^{4} \\epsilon^{2} + 8 P R Ug^{4} \\epsilon + 2 P R Ug^{4} - 16 P R Ug^{3} \\epsilon^{2} - 16 P R Ug^{3} \\epsilon - 4 P R Ug^{3} + 8 P R Ug^{2} \\epsilon^{2} + 8 P R Ug^{2} \\epsilon + 2 P R Ug^{2} - 8 P S Ug^{4} \\epsilon^{2} - 8 P S Ug^{4} \\epsilon - 2 P S Ug^{4} + 24 P S Ug^{3} \\epsilon^{2} + 24 P S Ug^{3} \\epsilon + 6 P S Ug^{3} - 24 P S Ug^{2} \\epsilon^{2} - 24 P S Ug^{2} \\epsilon - 6 P S Ug^{2} + 8 P S Ug \\epsilon^{2} + 8 P S Ug \\epsilon + 2 P S Ug - 8 P T Ug^{4} \\epsilon^{2} - 8 P T Ug^{4} \\epsilon - 2 P T Ug^{4} + 24 P T Ug^{3} \\epsilon^{2} + 24 P T Ug^{3} \\epsilon + 6 P T Ug^{3} - 24 P T Ug^{2} \\epsilon^{2} - 24 P T Ug^{2} \\epsilon - 6 P T Ug^{2} + 8 P T Ug \\epsilon^{2} + 8 P T Ug \\epsilon + 2 P T Ug + 4 R^{2} Ug^{4} \\epsilon^{2} + 4 R^{2} Ug^{4} \\epsilon + R^{2} Ug^{4} - 8 R S Ug^{4} \\epsilon^{2} - 8 R S Ug^{4} \\epsilon - 2 R S Ug^{4} + 8 R S Ug^{3} \\epsilon^{2} + 8 R S Ug^{3} \\epsilon + 2 R S Ug^{3} - 8 R T Ug^{4} \\epsilon^{2} - 8 R T Ug^{4} \\epsilon - 2 R T Ug^{4} + 8 R T Ug^{3} \\epsilon^{2} + 8 R T Ug^{3} \\epsilon + 2 R T Ug^{3} + 4 S^{2} Ug^{4} \\epsilon^{2} + 4 S^{2} Ug^{4} \\epsilon + S^{2} Ug^{4} - 8 S^{2} Ug^{3} \\epsilon^{2} - 8 S^{2} Ug^{3} \\epsilon - 2 S^{2} Ug^{3} + 4 S^{2} Ug^{2} \\epsilon^{2} + 4 S^{2} Ug^{2} \\epsilon + S^{2} Ug^{2} + 8 S T Ug^{4} \\epsilon^{2} + 8 S T Ug^{4} \\epsilon + 2 S T Ug^{4} - 16 S T Ug^{3} \\epsilon^{2} - 16 S T Ug^{3} \\epsilon - 4 S T Ug^{3} + 8 S T Ug^{2} \\epsilon^{2} + 8 S T Ug^{2} \\epsilon + 2 S T Ug^{2} + 4 T^{2} Ug^{4} \\epsilon^{2} + 4 T^{2} Ug^{4} \\epsilon + T^{2} Ug^{4} - 8 T^{2} Ug^{3} \\epsilon^{2} - 8 T^{2} Ug^{3} \\epsilon - 2 T^{2} Ug^{3} + 4 T^{2} Ug^{2} \\epsilon^{2} + 4 T^{2} Ug^{2} \\epsilon + T^{2} Ug^{2}}}{4 FG}$"
      ],
      "text/plain": [
       "-(3*FG + 2*P*Ug**2*epsilon - 3*P*Ug**2 - 4*P*Ug*epsilon + 6*P*Ug + 2*P*epsilon - 3*P + 2*R*Ug**2*epsilon - 3*R*Ug**2 - 2*S*Ug**2*epsilon + 3*S*Ug**2 + 2*S*Ug*epsilon - 3*S*Ug - 2*T*Ug**2*epsilon + 3*T*Ug**2 + 2*T*Ug*epsilon - 3*T*Ug + sqrt(FG**2 + 16*FG*P*Ug**2*epsilon**2 + 4*FG*P*Ug**2*epsilon - 2*FG*P*Ug**2 - 32*FG*P*Ug*epsilon**2 - 8*FG*P*Ug*epsilon + 4*FG*P*Ug + 16*FG*P*epsilon**2 + 4*FG*P*epsilon - 2*FG*P + 4*FG*R*Ug**2*epsilon + 2*FG*R*Ug**2 - 4*FG*S*Ug**2*epsilon - 2*FG*S*Ug**2 + 4*FG*S*Ug*epsilon + 2*FG*S*Ug - 16*FG*T*Ug**2*epsilon**2 - 4*FG*T*Ug**2*epsilon + 2*FG*T*Ug**2 + 16*FG*T*Ug*epsilon**2 + 4*FG*T*Ug*epsilon - 2*FG*T*Ug + 4*P**2*Ug**4*epsilon**2 + 4*P**2*Ug**4*epsilon + P**2*Ug**4 - 16*P**2*Ug**3*epsilon**2 - 16*P**2*Ug**3*epsilon - 4*P**2*Ug**3 + 24*P**2*Ug**2*epsilon**2 + 24*P**2*Ug**2*epsilon + 6*P**2*Ug**2 - 16*P**2*Ug*epsilon**2 - 16*P**2*Ug*epsilon - 4*P**2*Ug + 4*P**2*epsilon**2 + 4*P**2*epsilon + P**2 + 8*P*R*Ug**4*epsilon**2 + 8*P*R*Ug**4*epsilon + 2*P*R*Ug**4 - 16*P*R*Ug**3*epsilon**2 - 16*P*R*Ug**3*epsilon - 4*P*R*Ug**3 + 8*P*R*Ug**2*epsilon**2 + 8*P*R*Ug**2*epsilon + 2*P*R*Ug**2 - 8*P*S*Ug**4*epsilon**2 - 8*P*S*Ug**4*epsilon - 2*P*S*Ug**4 + 24*P*S*Ug**3*epsilon**2 + 24*P*S*Ug**3*epsilon + 6*P*S*Ug**3 - 24*P*S*Ug**2*epsilon**2 - 24*P*S*Ug**2*epsilon - 6*P*S*Ug**2 + 8*P*S*Ug*epsilon**2 + 8*P*S*Ug*epsilon + 2*P*S*Ug - 8*P*T*Ug**4*epsilon**2 - 8*P*T*Ug**4*epsilon - 2*P*T*Ug**4 + 24*P*T*Ug**3*epsilon**2 + 24*P*T*Ug**3*epsilon + 6*P*T*Ug**3 - 24*P*T*Ug**2*epsilon**2 - 24*P*T*Ug**2*epsilon - 6*P*T*Ug**2 + 8*P*T*Ug*epsilon**2 + 8*P*T*Ug*epsilon + 2*P*T*Ug + 4*R**2*Ug**4*epsilon**2 + 4*R**2*Ug**4*epsilon + R**2*Ug**4 - 8*R*S*Ug**4*epsilon**2 - 8*R*S*Ug**4*epsilon - 2*R*S*Ug**4 + 8*R*S*Ug**3*epsilon**2 + 8*R*S*Ug**3*epsilon + 2*R*S*Ug**3 - 8*R*T*Ug**4*epsilon**2 - 8*R*T*Ug**4*epsilon - 2*R*T*Ug**4 + 8*R*T*Ug**3*epsilon**2 + 8*R*T*Ug**3*epsilon + 2*R*T*Ug**3 + 4*S**2*Ug**4*epsilon**2 + 4*S**2*Ug**4*epsilon + S**2*Ug**4 - 8*S**2*Ug**3*epsilon**2 - 8*S**2*Ug**3*epsilon - 2*S**2*Ug**3 + 4*S**2*Ug**2*epsilon**2 + 4*S**2*Ug**2*epsilon + S**2*Ug**2 + 8*S*T*Ug**4*epsilon**2 + 8*S*T*Ug**4*epsilon + 2*S*T*Ug**4 - 16*S*T*Ug**3*epsilon**2 - 16*S*T*Ug**3*epsilon - 4*S*T*Ug**3 + 8*S*T*Ug**2*epsilon**2 + 8*S*T*Ug**2*epsilon + 2*S*T*Ug**2 + 4*T**2*Ug**4*epsilon**2 + 4*T**2*Ug**4*epsilon + T**2*Ug**4 - 8*T**2*Ug**3*epsilon**2 - 8*T**2*Ug**3*epsilon - 2*T**2*Ug**3 + 4*T**2*Ug**2*epsilon**2 + 4*T**2*Ug**2*epsilon + T**2*Ug**2))/(4*FG)"
      ]
     },
     "execution_count": 103,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e2.subs({(4*P**2*Ug**4 - 16*P**2*Ug**3 + 24*P**2*Ug**2 - 16*P**2*Ug + 4*P**2 + 8*P*R*Ug**4 - 16*P*R*Ug**3 + 8*P*R*Ug**2 - 8*P*S*Ug**4 + 24*P*S*Ug**3 - 24*P*S*Ug**2 + 8*P*S*Ug - 8*P*T*Ug**4 + 24*P*T*Ug**3 - 24*P*T*Ug**2 + 8*P*T*Ug + 4*R**2*Ug**4 - 8*R*S*Ug**4 + 8*R*S*Ug**3 - 8*R*T*Ug**4 + 8*R*T*Ug**3 + 4*S**2*Ug**4 - 8*S**2*Ug**3 + 4*S**2*Ug**2 + 8*S*T*Ug**4 - 16*S*T*Ug**3 + 8*S*T*Ug**2 + 4*T**2*Ug**4 - 8*T**2*Ug**3 + 4*T**2*Ug**2): Fg**2})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 145,
   "metadata": {},
   "outputs": [],
   "source": [
    "delta = (sym.factor(FG**2 + 16*FG*P*Ug**2*epsilon**2 + 4*FG*P*Ug**2*epsilon - 2*FG*P*Ug**2 - 32*FG*P*Ug*epsilon**2 - 8*FG*P*Ug*epsilon + 4*FG*P*Ug + 16*FG*P*epsilon**2 + 4*FG*P*epsilon - 2*FG*P + 4*FG*R*Ug**2*epsilon + 2*FG*R*Ug**2 - 4*FG*S*Ug**2*epsilon - 2*FG*S*Ug**2 + 4*FG*S*Ug*epsilon + 2*FG*S*Ug - 16*FG*T*Ug**2*epsilon**2 - 4*FG*T*Ug**2*epsilon + 2*FG*T*Ug**2 + 16*FG*T*Ug*epsilon**2 + 4*FG*T*Ug*epsilon - 2*FG*T*Ug + 4*P**2*Ug**4*epsilon**2 + 4*P**2*Ug**4*epsilon + P**2*Ug**4 - 16*P**2*Ug**3*epsilon**2 - 16*P**2*Ug**3*epsilon - 4*P**2*Ug**3 + 24*P**2*Ug**2*epsilon**2 + 24*P**2*Ug**2*epsilon + 6*P**2*Ug**2 - 16*P**2*Ug*epsilon**2 - 16*P**2*Ug*epsilon - 4*P**2*Ug + 4*P**2*epsilon**2 + 4*P**2*epsilon + P**2 + 8*P*R*Ug**4*epsilon**2 + 8*P*R*Ug**4*epsilon + 2*P*R*Ug**4 - 16*P*R*Ug**3*epsilon**2 - 16*P*R*Ug**3*epsilon - 4*P*R*Ug**3 + 8*P*R*Ug**2*epsilon**2 + 8*P*R*Ug**2*epsilon + 2*P*R*Ug**2 - 8*P*S*Ug**4*epsilon**2 - 8*P*S*Ug**4*epsilon - 2*P*S*Ug**4 + 24*P*S*Ug**3*epsilon**2 + 24*P*S*Ug**3*epsilon + 6*P*S*Ug**3 - 24*P*S*Ug**2*epsilon**2 - 24*P*S*Ug**2*epsilon - 6*P*S*Ug**2 + 8*P*S*Ug*epsilon**2 + 8*P*S*Ug*epsilon + 2*P*S*Ug - 8*P*T*Ug**4*epsilon**2 - 8*P*T*Ug**4*epsilon - 2*P*T*Ug**4 + 24*P*T*Ug**3*epsilon**2 + 24*P*T*Ug**3*epsilon + 6*P*T*Ug**3 - 24*P*T*Ug**2*epsilon**2 - 24*P*T*Ug**2*epsilon - 6*P*T*Ug**2 + 8*P*T*Ug*epsilon**2 + 8*P*T*Ug*epsilon + 2*P*T*Ug + 4*R**2*Ug**4*epsilon**2 + 4*R**2*Ug**4*epsilon + R**2*Ug**4 - 8*R*S*Ug**4*epsilon**2 - 8*R*S*Ug**4*epsilon - 2*R*S*Ug**4 + 8*R*S*Ug**3*epsilon**2 + 8*R*S*Ug**3*epsilon + 2*R*S*Ug**3 - 8*R*T*Ug**4*epsilon**2 - 8*R*T*Ug**4*epsilon - 2*R*T*Ug**4 + 8*R*T*Ug**3*epsilon**2 + 8*R*T*Ug**3*epsilon + 2*R*T*Ug**3 + 4*S**2*Ug**4*epsilon**2 + 4*S**2*Ug**4*epsilon + S**2*Ug**4 - 8*S**2*Ug**3*epsilon**2 - 8*S**2*Ug**3*epsilon - 2*S**2*Ug**3 + 4*S**2*Ug**2*epsilon**2 + 4*S**2*Ug**2*epsilon + S**2*Ug**2 + 8*S*T*Ug**4*epsilon**2 + 8*S*T*Ug**4*epsilon + 2*S*T*Ug**4 - 16*S*T*Ug**3*epsilon**2 - 16*S*T*Ug**3*epsilon - 4*S*T*Ug**3 + 8*S*T*Ug**2*epsilon**2 + 8*S*T*Ug**2*epsilon + 2*S*T*Ug**2 + 4*T**2*Ug**4*epsilon**2 + 4*T**2*Ug**4*epsilon + T**2*Ug**4 - 8*T**2*Ug**3*epsilon**2 - 8*T**2*Ug**3*epsilon - 2*T**2*Ug**3 + 4*T**2*Ug**2*epsilon**2 + 4*T**2*Ug**2*epsilon + T**2*Ug**2, epsilon, FG)\n",
    "    .subs({(4*P**2*Ug**4 - 16*P**2*Ug**3 + 24*P**2*Ug**2 - 16*P**2*Ug + 4*P**2 + 8*P*R*Ug**4 - 16*P*R*Ug**3 + 8*P*R*Ug**2 - 8*P*S*Ug**4 + 24*P*S*Ug**3 - 24*P*S*Ug**2 + 8*P*S*Ug - 8*P*T*Ug**4 + 24*P*T*Ug**3 - 24*P*T*Ug**2 + 8*P*T*Ug + 4*R**2*Ug**4 - 8*R*S*Ug**4 + 8*R*S*Ug**3 - 8*R*T*Ug**4 + 8*R*T*Ug**3 + 4*S**2*Ug**4 - 8*S**2*Ug**3 + 4*S**2*Ug**2 + 8*S*T*Ug**4 - 16*S*T*Ug**3 + 8*S*T*Ug**2 + 4*T**2*Ug**4 - 8*T**2*Ug**3 + 4*T**2*Ug**2): Fg**2})\n",
    "    .subs({P**2*Ug**4 - 4*P**2*Ug**3 + 6*P**2*Ug**2 - 4*P**2*Ug + P**2 + 2*P*R*Ug**4 - 4*P*R*Ug**3 + 2*P*R*Ug**2 - 2*P*S*Ug**4 + 6*P*S*Ug**3 - 6*P*S*Ug**2 + 2*P*S*Ug - 2*P*T*Ug**4 + 6*P*T*Ug**3 - 6*P*T*Ug**2 + 2*P*T*Ug + R**2*Ug**4 - 2*R*S*Ug**4 + 2*R*S*Ug**3 - 2*R*T*Ug**4 + 2*R*T*Ug**3 + S**2*Ug**4 - 2*S**2*Ug**3 + S**2*Ug**2 + 2*S*T*Ug**4 - 4*S*T*Ug**3 + 2*S*T*Ug**2 + T**2*Ug**4 - 2*T**2*Ug**3 + T**2*Ug**2: 0.25 * Fg**2})\n",
    "    .subs({4*P*Ug**2 - 8*P*Ug + 4*P + 4*R*Ug**2 - 4*S*Ug**2 + 4*S*Ug - 4*T*Ug**2 + 4*T*Ug: 2 * Fg})\n",
    "    .subs({-2*P*Ug**2 + 4*P*Ug - 2*P + 2*R*Ug**2 - 2*S*Ug**2 + 2*S*Ug + 2*T*Ug**2 - 2*T*Ug: -Fg + 4 * ((R - S) * Ug**2 + S * Ug)})\n",
    "    .subs({16*P*Ug**2 - 32*P*Ug + 16*P - 16*T*Ug**2 + 16*T*Ug: 8 * Fg - 16 * ((R - S) * Ug**2 + S * Ug)}))\n",
    "\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 16.0 \\left(0.0625 FG^{2} + 0.5 FG Fg \\epsilon^{2} + 0.125 FG Fg \\epsilon - 0.0625 FG Fg - 1.0 FG R Ug^{2} \\epsilon^{2} + 0.25 FG R Ug^{2} + 1.0 FG S Ug^{2} \\epsilon^{2} - 0.25 FG S Ug^{2} - 1.0 FG S Ug \\epsilon^{2} + 0.25 FG S Ug + 0.0625 Fg^{2} \\epsilon^{2} + 0.0625 Fg^{2} \\epsilon + 0.015625 Fg^{2}\\right)$"
      ],
      "text/plain": [
       "16.0*(0.0625*FG**2 + 0.5*FG*Fg*epsilon**2 + 0.125*FG*Fg*epsilon - 0.0625*FG*Fg - 1.0*FG*R*Ug**2*epsilon**2 + 0.25*FG*R*Ug**2 + 1.0*FG*S*Ug**2*epsilon**2 - 0.25*FG*S*Ug**2 - 1.0*FG*S*Ug*epsilon**2 + 0.25*FG*S*Ug + 0.0625*Fg**2*epsilon**2 + 0.0625*Fg**2*epsilon + 0.015625*Fg**2)"
      ]
     },
     "execution_count": 146,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(delta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 155,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle FG^{2} + FG \\left(- Fg + 4 S Ug + 4 Ug^{2} \\left(R - S\\right)\\right) + 0.25 Fg^{2}$"
      ],
      "text/plain": [
       "FG**2 + FG*(-Fg + 4*S*Ug + 4*Ug**2*(R - S)) + 0.25*Fg**2"
      ]
     },
     "execution_count": 155,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "delta.subs({epsilon: 0})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 72.0 \\left(0.111111111111111 FG^{2} - 0.111111111111111 FG Fg \\epsilon^{2} + 0.0555555555555556 FG Fg \\epsilon - 0.111111111111111 FG Fg + FG \\epsilon^{2} \\left(0.222222222222222 R Ug^{2} - 0.222222222222222 S Ug^{2} + 0.222222222222222 S Ug\\right) + FG \\left(0.444444444444444 R Ug^{2} - 0.444444444444444 S Ug^{2} + 0.444444444444444 S Ug\\right) - 0.0555555555555556 Fg^{2} \\epsilon + 0.0277777777777778 Fg^{2} + Fg \\epsilon \\left(0.166666666666667 R Ug^{2} - 0.166666666666667 S Ug^{2} + 0.166666666666667 S Ug\\right) + Fg \\left(- 0.25 R Ug^{2} + 0.25 S Ug^{2} - 0.25 S Ug\\right) + 0.5 R^{2} Ug^{4} - 1.0 R S Ug^{4} + 1.0 R S Ug^{3} + 0.5 S^{2} Ug^{4} - 1.0 S^{2} Ug^{3} + 0.5 S^{2} Ug^{2}\\right)$"
      ],
      "text/plain": [
       "-72.0*(0.111111111111111*FG**2 - 0.111111111111111*FG*Fg*epsilon**2 + 0.0555555555555556*FG*Fg*epsilon - 0.111111111111111*FG*Fg + FG*epsilon**2*(0.222222222222222*R*Ug**2 - 0.222222222222222*S*Ug**2 + 0.222222222222222*S*Ug) + FG*(0.444444444444444*R*Ug**2 - 0.444444444444444*S*Ug**2 + 0.444444444444444*S*Ug) - 0.0555555555555556*Fg**2*epsilon + 0.0277777777777778*Fg**2 + Fg*epsilon*(0.166666666666667*R*Ug**2 - 0.166666666666667*S*Ug**2 + 0.166666666666667*S*Ug) + Fg*(-0.25*R*Ug**2 + 0.25*S*Ug**2 - 0.25*S*Ug) + 0.5*R**2*Ug**4 - 1.0*R*S*Ug**4 + 1.0*R*S*Ug**3 + 0.5*S**2*Ug**4 - 1.0*S**2*Ug**3 + 0.5*S**2*Ug**2)"
      ]
     },
     "execution_count": 153,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(delta - r**2, FG, Fg, epsilon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 121,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 32.0 \\left(0.03125 FG^{2} + 0.0625 FG Fg \\epsilon - 0.03125 FG Fg + 0.5 FG P Ug^{2} \\epsilon^{2} - 1.0 FG P Ug \\epsilon^{2} + 0.5 FG P \\epsilon^{2} + 0.125 FG R Ug^{2} - 0.125 FG S Ug^{2} + 0.125 FG S Ug - 0.5 FG T Ug^{2} \\epsilon^{2} + 0.5 FG T Ug \\epsilon^{2} + 0.03125 Fg^{2} \\epsilon^{2} + 0.03125 Fg^{2} \\epsilon + 0.015625 Fg\\right)$"
      ],
      "text/plain": [
       "32.0*(0.03125*FG**2 + 0.0625*FG*Fg*epsilon - 0.03125*FG*Fg + 0.5*FG*P*Ug**2*epsilon**2 - 1.0*FG*P*Ug*epsilon**2 + 0.5*FG*P*epsilon**2 + 0.125*FG*R*Ug**2 - 0.125*FG*S*Ug**2 + 0.125*FG*S*Ug - 0.5*FG*T*Ug**2*epsilon**2 + 0.5*FG*T*Ug*epsilon**2 + 0.03125*Fg**2*epsilon**2 + 0.03125*Fg**2*epsilon + 0.015625*Fg)"
      ]
     },
     "execution_count": 121,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(delta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 101,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 4 \\left(P Ug^{2} - 2 P Ug + P + R Ug^{2} - S Ug^{2} + S Ug - T Ug^{2} + T Ug\\right)^{2}$"
      ],
      "text/plain": [
       "4*(P*Ug**2 - 2*P*Ug + P + R*Ug**2 - S*Ug**2 + S*Ug - T*Ug**2 + T*Ug)**2"
      ]
     },
     "execution_count": 101,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor((4*P**2*Ug**4 - 16*P**2*Ug**3 + 24*P**2*Ug**2 - 16*P**2*Ug + 4*P**2 + 8*P*R*Ug**4 - 16*P*R*Ug**3 + 8*P*R*Ug**2 - 8*P*S*Ug**4 + 24*P*S*Ug**3 - 24*P*S*Ug**2 + 8*P*S*Ug - 8*P*T*Ug**4 + 24*P*T*Ug**3 - 24*P*T*Ug**2 + 8*P*T*Ug + 4*R**2*Ug**4 - 8*R*S*Ug**4 + 8*R*S*Ug**3 - 8*R*T*Ug**4 + 8*R*T*Ug**3 + 4*S**2*Ug**4 - 8*S**2*Ug**3 + 4*S**2*Ug**2 + 8*S*T*Ug**4 - 16*S*T*Ug**3 + 8*S*T*Ug**2 + 4*T**2*Ug**4 - 8*T**2*Ug**3 + 4*T**2*Ug**2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 111,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(P Ug^{2} - 2 P Ug + P + R Ug^{2} - S Ug^{2} + S Ug - T Ug^{2} + T Ug\\right)^{2}$"
      ],
      "text/plain": [
       "(P*Ug**2 - 2*P*Ug + P + R*Ug**2 - S*Ug**2 + S*Ug - T*Ug**2 + T*Ug)**2"
      ]
     },
     "execution_count": 111,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(P**2*Ug**4 - 4*P**2*Ug**3 + 6*P**2*Ug**2 - 4*P**2*Ug + P**2 + 2*P*R*Ug**4 - 4*P*R*Ug**3 + 2*P*R*Ug**2 - 2*P*S*Ug**4 + 6*P*S*Ug**3 - 6*P*S*Ug**2 + 2*P*S*Ug - 2*P*T*Ug**4 + 6*P*T*Ug**3 - 6*P*T*Ug**2 + 2*P*T*Ug + R**2*Ug**4 - 2*R*S*Ug**4 + 2*R*S*Ug**3 - 2*R*T*Ug**4 + 2*R*T*Ug**3 + S**2*Ug**4 - 2*S**2*Ug**3 + S**2*Ug**2 + 2*S*T*Ug**4 - 4*S*T*Ug**3 + 2*S*T*Ug**2 + T**2*Ug**4 - 2*T**2*Ug**3 + T**2*Ug**2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 115,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 4 \\left(P Ug^{2} - 2 P Ug + P + R Ug^{2} - S Ug^{2} + S Ug - T Ug^{2} + T Ug\\right)$"
      ],
      "text/plain": [
       "4*(P*Ug**2 - 2*P*Ug + P + R*Ug**2 - S*Ug**2 + S*Ug - T*Ug**2 + T*Ug)"
      ]
     },
     "execution_count": 115,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(4*P*Ug**2 - 8*P*Ug + 4*P + 4*R*Ug**2 - 4*S*Ug**2 + 4*S*Ug - 4*T*Ug**2 + 4*T*Ug)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 117,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - 2 \\left(P Ug^{2} - 2 P Ug + P - R Ug^{2} + S Ug^{2} - S Ug - T Ug^{2} + T Ug\\right)$"
      ],
      "text/plain": [
       "-2*(P*Ug**2 - 2*P*Ug + P - R*Ug**2 + S*Ug**2 - S*Ug - T*Ug**2 + T*Ug)"
      ]
     },
     "execution_count": 117,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(-2*P*Ug**2 + 4*P*Ug - 2*P + 2*R*Ug**2 - 2*S*Ug**2 + 2*S*Ug + 2*T*Ug**2 - 2*T*Ug)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 119,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle 16 \\left(Ug - 1\\right) \\left(P Ug - P - T Ug\\right)$"
      ],
      "text/plain": [
       "16*(Ug - 1)*(P*Ug - P - T*Ug)"
      ]
     },
     "execution_count": 119,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(16*P*Ug**2 - 32*P*Ug + 16*P - 16*T*Ug**2 + 16*T*Ug)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 93,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle - \\frac{3 FG + 2 P Ug^{2} \\epsilon - 3 P Ug^{2} - 4 P Ug \\epsilon + 6 P Ug + 2 P \\epsilon - 3 P + 2 R Ug^{2} \\epsilon - 3 R Ug^{2} - 2 S Ug^{2} \\epsilon + 3 S Ug^{2} + 2 S Ug \\epsilon - 3 S Ug - 2 T Ug^{2} \\epsilon + 3 T Ug^{2} + 2 T Ug \\epsilon - 3 T Ug - \\sqrt{FG^{2} + 16 FG P Ug^{2} \\epsilon^{2} + 4 FG P Ug^{2} \\epsilon - 2 FG P Ug^{2} - 32 FG P Ug \\epsilon^{2} - 8 FG P Ug \\epsilon + 4 FG P Ug + 16 FG P \\epsilon^{2} + 4 FG P \\epsilon - 2 FG P + 4 FG R Ug^{2} \\epsilon + 2 FG R Ug^{2} - 4 FG S Ug^{2} \\epsilon - 2 FG S Ug^{2} + 4 FG S Ug \\epsilon + 2 FG S Ug - 16 FG T Ug^{2} \\epsilon^{2} - 4 FG T Ug^{2} \\epsilon + 2 FG T Ug^{2} + 16 FG T Ug \\epsilon^{2} + 4 FG T Ug \\epsilon - 2 FG T Ug + 4 P^{2} Ug^{4} \\epsilon^{2} + 4 P^{2} Ug^{4} \\epsilon + P^{2} Ug^{4} - 16 P^{2} Ug^{3} \\epsilon^{2} - 16 P^{2} Ug^{3} \\epsilon - 4 P^{2} Ug^{3} + 24 P^{2} Ug^{2} \\epsilon^{2} + 24 P^{2} Ug^{2} \\epsilon + 6 P^{2} Ug^{2} - 16 P^{2} Ug \\epsilon^{2} - 16 P^{2} Ug \\epsilon - 4 P^{2} Ug + 4 P^{2} \\epsilon^{2} + 4 P^{2} \\epsilon + P^{2} + 8 P R Ug^{4} \\epsilon^{2} + 8 P R Ug^{4} \\epsilon + 2 P R Ug^{4} - 16 P R Ug^{3} \\epsilon^{2} - 16 P R Ug^{3} \\epsilon - 4 P R Ug^{3} + 8 P R Ug^{2} \\epsilon^{2} + 8 P R Ug^{2} \\epsilon + 2 P R Ug^{2} - 8 P S Ug^{4} \\epsilon^{2} - 8 P S Ug^{4} \\epsilon - 2 P S Ug^{4} + 24 P S Ug^{3} \\epsilon^{2} + 24 P S Ug^{3} \\epsilon + 6 P S Ug^{3} - 24 P S Ug^{2} \\epsilon^{2} - 24 P S Ug^{2} \\epsilon - 6 P S Ug^{2} + 8 P S Ug \\epsilon^{2} + 8 P S Ug \\epsilon + 2 P S Ug - 8 P T Ug^{4} \\epsilon^{2} - 8 P T Ug^{4} \\epsilon - 2 P T Ug^{4} + 24 P T Ug^{3} \\epsilon^{2} + 24 P T Ug^{3} \\epsilon + 6 P T Ug^{3} - 24 P T Ug^{2} \\epsilon^{2} - 24 P T Ug^{2} \\epsilon - 6 P T Ug^{2} + 8 P T Ug \\epsilon^{2} + 8 P T Ug \\epsilon + 2 P T Ug + 4 R^{2} Ug^{4} \\epsilon^{2} + 4 R^{2} Ug^{4} \\epsilon + R^{2} Ug^{4} - 8 R S Ug^{4} \\epsilon^{2} - 8 R S Ug^{4} \\epsilon - 2 R S Ug^{4} + 8 R S Ug^{3} \\epsilon^{2} + 8 R S Ug^{3} \\epsilon + 2 R S Ug^{3} - 8 R T Ug^{4} \\epsilon^{2} - 8 R T Ug^{4} \\epsilon - 2 R T Ug^{4} + 8 R T Ug^{3} \\epsilon^{2} + 8 R T Ug^{3} \\epsilon + 2 R T Ug^{3} + 4 S^{2} Ug^{4} \\epsilon^{2} + 4 S^{2} Ug^{4} \\epsilon + S^{2} Ug^{4} - 8 S^{2} Ug^{3} \\epsilon^{2} - 8 S^{2} Ug^{3} \\epsilon - 2 S^{2} Ug^{3} + 4 S^{2} Ug^{2} \\epsilon^{2} + 4 S^{2} Ug^{2} \\epsilon + S^{2} Ug^{2} + 8 S T Ug^{4} \\epsilon^{2} + 8 S T Ug^{4} \\epsilon + 2 S T Ug^{4} - 16 S T Ug^{3} \\epsilon^{2} - 16 S T Ug^{3} \\epsilon - 4 S T Ug^{3} + 8 S T Ug^{2} \\epsilon^{2} + 8 S T Ug^{2} \\epsilon + 2 S T Ug^{2} + 4 T^{2} Ug^{4} \\epsilon^{2} + 4 T^{2} Ug^{4} \\epsilon + T^{2} Ug^{4} - 8 T^{2} Ug^{3} \\epsilon^{2} - 8 T^{2} Ug^{3} \\epsilon - 2 T^{2} Ug^{3} + 4 T^{2} Ug^{2} \\epsilon^{2} + 4 T^{2} Ug^{2} \\epsilon + T^{2} Ug^{2}}}{4 FG}$"
      ],
      "text/plain": [
       "-(3*FG + 2*P*Ug**2*epsilon - 3*P*Ug**2 - 4*P*Ug*epsilon + 6*P*Ug + 2*P*epsilon - 3*P + 2*R*Ug**2*epsilon - 3*R*Ug**2 - 2*S*Ug**2*epsilon + 3*S*Ug**2 + 2*S*Ug*epsilon - 3*S*Ug - 2*T*Ug**2*epsilon + 3*T*Ug**2 + 2*T*Ug*epsilon - 3*T*Ug - sqrt(FG**2 + 16*FG*P*Ug**2*epsilon**2 + 4*FG*P*Ug**2*epsilon - 2*FG*P*Ug**2 - 32*FG*P*Ug*epsilon**2 - 8*FG*P*Ug*epsilon + 4*FG*P*Ug + 16*FG*P*epsilon**2 + 4*FG*P*epsilon - 2*FG*P + 4*FG*R*Ug**2*epsilon + 2*FG*R*Ug**2 - 4*FG*S*Ug**2*epsilon - 2*FG*S*Ug**2 + 4*FG*S*Ug*epsilon + 2*FG*S*Ug - 16*FG*T*Ug**2*epsilon**2 - 4*FG*T*Ug**2*epsilon + 2*FG*T*Ug**2 + 16*FG*T*Ug*epsilon**2 + 4*FG*T*Ug*epsilon - 2*FG*T*Ug + 4*P**2*Ug**4*epsilon**2 + 4*P**2*Ug**4*epsilon + P**2*Ug**4 - 16*P**2*Ug**3*epsilon**2 - 16*P**2*Ug**3*epsilon - 4*P**2*Ug**3 + 24*P**2*Ug**2*epsilon**2 + 24*P**2*Ug**2*epsilon + 6*P**2*Ug**2 - 16*P**2*Ug*epsilon**2 - 16*P**2*Ug*epsilon - 4*P**2*Ug + 4*P**2*epsilon**2 + 4*P**2*epsilon + P**2 + 8*P*R*Ug**4*epsilon**2 + 8*P*R*Ug**4*epsilon + 2*P*R*Ug**4 - 16*P*R*Ug**3*epsilon**2 - 16*P*R*Ug**3*epsilon - 4*P*R*Ug**3 + 8*P*R*Ug**2*epsilon**2 + 8*P*R*Ug**2*epsilon + 2*P*R*Ug**2 - 8*P*S*Ug**4*epsilon**2 - 8*P*S*Ug**4*epsilon - 2*P*S*Ug**4 + 24*P*S*Ug**3*epsilon**2 + 24*P*S*Ug**3*epsilon + 6*P*S*Ug**3 - 24*P*S*Ug**2*epsilon**2 - 24*P*S*Ug**2*epsilon - 6*P*S*Ug**2 + 8*P*S*Ug*epsilon**2 + 8*P*S*Ug*epsilon + 2*P*S*Ug - 8*P*T*Ug**4*epsilon**2 - 8*P*T*Ug**4*epsilon - 2*P*T*Ug**4 + 24*P*T*Ug**3*epsilon**2 + 24*P*T*Ug**3*epsilon + 6*P*T*Ug**3 - 24*P*T*Ug**2*epsilon**2 - 24*P*T*Ug**2*epsilon - 6*P*T*Ug**2 + 8*P*T*Ug*epsilon**2 + 8*P*T*Ug*epsilon + 2*P*T*Ug + 4*R**2*Ug**4*epsilon**2 + 4*R**2*Ug**4*epsilon + R**2*Ug**4 - 8*R*S*Ug**4*epsilon**2 - 8*R*S*Ug**4*epsilon - 2*R*S*Ug**4 + 8*R*S*Ug**3*epsilon**2 + 8*R*S*Ug**3*epsilon + 2*R*S*Ug**3 - 8*R*T*Ug**4*epsilon**2 - 8*R*T*Ug**4*epsilon - 2*R*T*Ug**4 + 8*R*T*Ug**3*epsilon**2 + 8*R*T*Ug**3*epsilon + 2*R*T*Ug**3 + 4*S**2*Ug**4*epsilon**2 + 4*S**2*Ug**4*epsilon + S**2*Ug**4 - 8*S**2*Ug**3*epsilon**2 - 8*S**2*Ug**3*epsilon - 2*S**2*Ug**3 + 4*S**2*Ug**2*epsilon**2 + 4*S**2*Ug**2*epsilon + S**2*Ug**2 + 8*S*T*Ug**4*epsilon**2 + 8*S*T*Ug**4*epsilon + 2*S*T*Ug**4 - 16*S*T*Ug**3*epsilon**2 - 16*S*T*Ug**3*epsilon - 4*S*T*Ug**3 + 8*S*T*Ug**2*epsilon**2 + 8*S*T*Ug**2*epsilon + 2*S*T*Ug**2 + 4*T**2*Ug**4*epsilon**2 + 4*T**2*Ug**4*epsilon + T**2*Ug**4 - 8*T**2*Ug**3*epsilon**2 - 8*T**2*Ug**3*epsilon - 2*T**2*Ug**3 + 4*T**2*Ug**2*epsilon**2 + 4*T**2*Ug**2*epsilon + T**2*Ug**2))/(4*FG)"
      ]
     },
     "execution_count": 93,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "e3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 133,
   "metadata": {},
   "outputs": [],
   "source": [
    "f1 = (sym.factor(sym.together(f1), UGA(x1 + x3), UgA(x1 + x3))\n",
    "         .subs({-0.5*M*x1 - 0.5*M*x2 + 0.5*P + (-1.0*P*x1 - 1.0*P*x2 + 0.5*S*x1 + 0.5*S*x2 + 0.5*T*x1 + 0.5*T*x2)*UGA(x1 + x3) + (0.5*P*x1 + 0.5*P*x2 + 0.5*R*x1 + 0.5*R*x2 - 0.5*S*x1 - 0.5*S*x2 - 0.5*T*x1 - 0.5*T*x2)*UGA(x1 + x3)**2 + (1.0*P*x1 + 1.0*P*x2 - 1.0*P - 0.5*S*x1 - 0.5*S*x2 + 0.5*S - 0.5*T*x1 - 0.5*T*x2 + 0.5*T)*UgA(x1 + x3) + (-0.5*P*x1 - 0.5*P*x2 + 0.5*P - 0.5*R*x1 - 0.5*R*x2 + 0.5*R + 0.5*S*x1 + 0.5*S*x2 - 0.5*S + 0.5*T*x1 + 0.5*T*x2 - 0.5*T)*UgA(x1 + x3)**2: N}))\n",
    "f2 = (sym.factor(sym.together(f2), UGA(x1 + x3), UgA(x1 + x3))\n",
    "         .subs({-0.5*M*x1 - 0.5*M*x2 + 0.5*P + (-1.0*P*x1 - 1.0*P*x2 + 0.5*S*x1 + 0.5*S*x2 + 0.5*T*x1 + 0.5*T*x2)*UGA(x1 + x3) + (0.5*P*x1 + 0.5*P*x2 + 0.5*R*x1 + 0.5*R*x2 - 0.5*S*x1 - 0.5*S*x2 - 0.5*T*x1 - 0.5*T*x2)*UGA(x1 + x3)**2 + (1.0*P*x1 + 1.0*P*x2 - 1.0*P - 0.5*S*x1 - 0.5*S*x2 + 0.5*S - 0.5*T*x1 - 0.5*T*x2 + 0.5*T)*UgA(x1 + x3) + (-0.5*P*x1 - 0.5*P*x2 + 0.5*P - 0.5*R*x1 - 0.5*R*x2 + 0.5*R + 0.5*S*x1 + 0.5*S*x2 - 0.5*S + 0.5*T*x1 + 0.5*T*x2 - 0.5*T)*UgA(x1 + x3)**2: N}))\n",
    "f3 = (sym.factor(sym.together(f3), UGA(x1 + x3), UgA(x1 + x3))\n",
    "         .subs({-0.5*M*x1 - 0.5*M*x2 + 0.5*P + (-1.0*P*x1 - 1.0*P*x2 + 0.5*S*x1 + 0.5*S*x2 + 0.5*T*x1 + 0.5*T*x2)*UGA(x1 + x3) + (0.5*P*x1 + 0.5*P*x2 + 0.5*R*x1 + 0.5*R*x2 - 0.5*S*x1 - 0.5*S*x2 - 0.5*T*x1 - 0.5*T*x2)*UGA(x1 + x3)**2 + (1.0*P*x1 + 1.0*P*x2 - 1.0*P - 0.5*S*x1 - 0.5*S*x2 + 0.5*S - 0.5*T*x1 - 0.5*T*x2 + 0.5*T)*UgA(x1 + x3) + (-0.5*P*x1 - 0.5*P*x2 + 0.5*P - 0.5*R*x1 - 0.5*R*x2 + 0.5*R + 0.5*S*x1 + 0.5*S*x2 - 0.5*S + 0.5*T*x1 + 0.5*T*x2 - 0.5*T)*UgA(x1 + x3)**2: N}))\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 131,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = sym.function('N')\n",
    "f1 = f1.subs({-0.5*M*x1 - 0.5*M*x2 + 0.5*P + (-1.0*P*x1 - 1.0*P*x2 + 0.5*S*x1 + 0.5*S*x2 + 0.5*T*x1 + 0.5*T*x2)*UGA(x1 + x3) + (0.5*P*x1 + 0.5*P*x2 + 0.5*R*x1 + 0.5*R*x2 - 0.5*S*x1 - 0.5*S*x2 - 0.5*T*x1 - 0.5*T*x2)*UGA(x1 + x3)**2 + (1.0*P*x1 + 1.0*P*x2 - 1.0*P - 0.5*S*x1 - 0.5*S*x2 + 0.5*S - 0.5*T*x1 - 0.5*T*x2 + 0.5*T)*UgA(x1 + x3) + (-0.5*P*x1 - 0.5*P*x2 + 0.5*P - 0.5*R*x1 - 0.5*R*x2 + 0.5*R + 0.5*S*x1 + 0.5*S*x2 - 0.5*S + 0.5*T*x1 + 0.5*T*x2 - 0.5*T)*UgA(x1 + x3)**2: N})\n",
    "f2 = f2.subs({-0.5*M*x1 - 0.5*M*x2 + 0.5*P + (-1.0*P*x1 - 1.0*P*x2 + 0.5*S*x1 + 0.5*S*x2 + 0.5*T*x1 + 0.5*T*x2)*UGA(x1 + x3) + (0.5*P*x1 + 0.5*P*x2 + 0.5*R*x1 + 0.5*R*x2 - 0.5*S*x1 - 0.5*S*x2 - 0.5*T*x1 - 0.5*T*x2)*UGA(x1 + x3)**2 + (1.0*P*x1 + 1.0*P*x2 - 1.0*P - 0.5*S*x1 - 0.5*S*x2 + 0.5*S - 0.5*T*x1 - 0.5*T*x2 + 0.5*T)*UgA(x1 + x3) + (-0.5*P*x1 - 0.5*P*x2 + 0.5*P - 0.5*R*x1 - 0.5*R*x2 + 0.5*R + 0.5*S*x1 + 0.5*S*x2 - 0.5*S + 0.5*T*x1 + 0.5*T*x2 - 0.5*T)*UgA(x1 + x3)**2: N})\n",
    "f3 = f3.subs({-0.5*M*x1 - 0.5*M*x2 + 0.5*P + (-1.0*P*x1 - 1.0*P*x2 + 0.5*S*x1 + 0.5*S*x2 + 0.5*T*x1 + 0.5*T*x2)*UGA(x1 + x3) + (0.5*P*x1 + 0.5*P*x2 + 0.5*R*x1 + 0.5*R*x2 - 0.5*S*x1 - 0.5*S*x2 - 0.5*T*x1 - 0.5*T*x2)*UGA(x1 + x3)**2 + (1.0*P*x1 + 1.0*P*x2 - 1.0*P - 0.5*S*x1 - 0.5*S*x2 + 0.5*S - 0.5*T*x1 - 0.5*T*x2 + 0.5*T)*UgA(x1 + x3) + (-0.5*P*x1 - 0.5*P*x2 + 0.5*P - 0.5*R*x1 - 0.5*R*x2 + 0.5*R + 0.5*S*x1 + 0.5*S*x2 - 0.5*S + 0.5*T*x1 + 0.5*T*x2 - 0.5*T)*UgA(x1 + x3)**2: N})\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {},
   "outputs": [],
   "source": [
    "sym.function?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 108,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-3.0*(-0.0833333333333333*M*epsilon*x2**2*x3 + 0.0833333333333333*M*epsilon*x2*x3**2 - 0.0833333333333333*M*epsilon*x2*x3 - 0.166666666666667*M*x1**4 - 0.166666666666667*P*epsilon*x2*x3**2 + 0.166666666666667*P*epsilon*x2*x3 + 0.0416666666666667*P*x2*x3**2 + x1**4*(-0.333333333333333*P + 0.166666666666667*S + 0.166666666666667*T)*UGA(x1 + x3) + x1**4*(0.333333333333333*P - 0.166666666666667*S - 0.166666666666667*T)*UgA(x1 + x3) + x1**4*(-0.166666666666667*P - 0.166666666666667*R + 0.166666666666667*S + 0.166666666666667*T)*UgA(x1 + x3)**2 + x1**4*(0.166666666666667*P + 0.166666666666667*R - 0.166666666666667*S - 0.166666666666667*T)*UGA(x1 + x3)**2 + x1**3*(-0.166666666666667*M*x2 - 0.333333333333333*M*x3 + 0.208333333333333*M + 0.125*P) + x1**3*(0.333333333333333*P*x2 + 0.666666666666667*P*x3 - 0.666666666666667*P - 0.166666666666667*S*x2 - 0.333333333333333*S*x3 + 0.375*S - 0.166666666666667*T*x2 - 0.333333333333333*T*x3 + 0.333333333333333*T)*UgA(x1 + x3) + x1**3*(-0.166666666666667*M*epsilon + 0.125*M - 0.333333333333333*P*x2 - 0.666666666666667*P*x3 + 0.416666666666667*P + 0.166666666666667*S*epsilon + 0.166666666666667*S*x2 + 0.333333333333333*S*x3 - 0.333333333333333*S + 0.166666666666667*T*x2 + 0.333333333333333*T*x3 - 0.208333333333333*T)*UGA(x1 + x3) + x1**3*(-0.166666666666667*P*x2 - 0.333333333333333*P*x3 + 0.333333333333333*P - 0.166666666666667*R*x2 - 0.333333333333333*R*x3 + 0.375*R + 0.166666666666667*S*x2 + 0.333333333333333*S*x3 - 0.375*S + 0.166666666666667*T*x2 + 0.333333333333333*T*x3 - 0.333333333333333*T)*UgA(x1 + x3)**2 + x1**3*(0.166666666666667*P*x2 + 0.333333333333333*P*x3 - 0.208333333333333*P + 0.166666666666667*R*epsilon + 0.166666666666667*R*x2 + 0.333333333333333*R*x3 - 0.333333333333333*R - 0.166666666666667*S*epsilon - 0.166666666666667*S*x2 - 0.333333333333333*S*x3 + 0.333333333333333*S - 0.166666666666667*T*x2 - 0.333333333333333*T*x3 + 0.208333333333333*T)*UGA(x1 + x3)**2 + x1**2*(0.0833333333333333*M*epsilon*x2 - 0.333333333333333*M*x2*x3 + 0.125*M*x2 - 0.166666666666667*M*x3**2 + 0.25*M*x3 - 0.0416666666666667*M - 0.166666666666667*P*epsilon*x2 + 0.0416666666666667*P*x2 + 0.25*P*x3 - 0.125*P) + x1**2*(0.166666666666667*P*epsilon*x2 + 0.666666666666667*P*x2*x3 - 0.333333333333333*P*x2 + 0.333333333333333*P*x3**2 - 1.0*P*x3 + 0.333333333333333*P + 0.0833333333333333*S*epsilon*x3 - 0.333333333333333*S*x2*x3 + 0.208333333333333*S*x2 - 0.166666666666667*S*x3**2 + 0.5*S*x3 - 0.25*S - 0.0833333333333333*T*epsilon*x2 - 0.333333333333333*T*x2*x3 + 0.166666666666667*T*x2 - 0.166666666666667*T*x3**2 + 0.5*T*x3 - 0.166666666666667*T)*UgA(x1 + x3) + x1**2*(-0.0833333333333333*P*epsilon*x2 - 0.333333333333333*P*x2*x3 + 0.166666666666667*P*x2 - 0.166666666666667*P*x3**2 + 0.5*P*x3 - 0.166666666666667*P + 0.0833333333333333*R*epsilon*x3 - 0.333333333333333*R*x2*x3 + 0.208333333333333*R*x2 - 0.166666666666667*R*x3**2 + 0.5*R*x3 - 0.25*R - 0.0833333333333333*S*epsilon*x3 + 0.333333333333333*S*x2*x3 - 0.208333333333333*S*x2 + 0.166666666666667*S*x3**2 - 0.5*S*x3 + 0.25*S + 0.0833333333333333*T*epsilon*x2 + 0.333333333333333*T*x2*x3 - 0.166666666666667*T*x2 + 0.166666666666667*T*x3**2 - 0.5*T*x3 + 0.166666666666667*T)*UgA(x1 + x3)**2 + x1**2*(-0.0833333333333333*M*epsilon*x2 - 0.25*M*epsilon*x3 + 0.166666666666667*M*epsilon + 0.125*M*x2 + 0.166666666666667*M*x3 - 0.125*M + 0.166666666666667*P*epsilon*x2 - 0.666666666666667*P*x2*x3 + 0.25*P*x2 - 0.333333333333333*P*x3**2 + 0.5*P*x3 - 0.0833333333333333*P + 0.25*S*epsilon*x3 - 0.166666666666667*S*epsilon + 0.333333333333333*S*x2*x3 - 0.25*S*x2 + 0.166666666666667*S*x3**2 - 0.416666666666667*S*x3 + 0.166666666666667*S - 0.0833333333333333*T*epsilon*x2 + 0.333333333333333*T*x2*x3 - 0.125*T*x2 + 0.166666666666667*T*x3**2 - 0.25*T*x3 + 0.0416666666666667*T)*UGA(x1 + x3) + x1**2*(-0.0833333333333333*P*epsilon*x2 + 0.333333333333333*P*x2*x3 - 0.125*P*x2 + 0.166666666666667*P*x3**2 - 0.25*P*x3 + 0.0416666666666667*P + 0.25*R*epsilon*x3 - 0.166666666666667*R*epsilon + 0.333333333333333*R*x2*x3 - 0.25*R*x2 + 0.166666666666667*R*x3**2 - 0.416666666666667*R*x3 + 0.166666666666667*R - 0.25*S*epsilon*x3 + 0.166666666666667*S*epsilon - 0.333333333333333*S*x2*x3 + 0.25*S*x2 - 0.166666666666667*S*x3**2 + 0.416666666666667*S*x3 - 0.166666666666667*S + 0.0833333333333333*T*epsilon*x2 - 0.333333333333333*T*x2*x3 + 0.125*T*x2 - 0.166666666666667*T*x3**2 + 0.25*T*x3 - 0.0416666666666667*T)*UGA(x1 + x3)**2 + x1*(-0.0833333333333333*M*epsilon*x2**2 + 0.166666666666667*M*epsilon*x2*x3 - 0.0833333333333333*M*epsilon*x2 - 0.166666666666667*M*x2*x3**2 + 0.125*M*x2*x3 + 0.0416666666666667*M*x3**2 - 0.0416666666666667*M*x3 - 0.333333333333333*P*epsilon*x2*x3 + 0.166666666666667*P*epsilon*x2 + 0.0833333333333333*P*x2*x3 + 0.125*P*x3**2 - 0.125*P*x3) + x1*(0.166666666666667*P*epsilon*x2**2 + 0.333333333333333*P*epsilon*x2*x3 - 0.166666666666667*P*epsilon*x2 + 0.333333333333333*P*x2*x3**2 - 0.416666666666667*P*x2*x3 - 0.333333333333333*P*x3**2 + 0.333333333333333*P*x3 + 0.0833333333333333*S*epsilon*x3**2 - 0.0833333333333333*S*epsilon*x3 - 0.166666666666667*S*x2*x3**2 + 0.208333333333333*S*x2*x3 - 0.0416666666666667*S*x2 + 0.125*S*x3**2 - 0.166666666666667*S*x3 + 0.0416666666666667*S - 0.0833333333333333*T*epsilon*x2**2 - 0.166666666666667*T*epsilon*x2*x3 + 0.0833333333333333*T*epsilon*x2 - 0.166666666666667*T*x2*x3**2 + 0.208333333333333*T*x2*x3 + 0.166666666666667*T*x3**2 - 0.166666666666667*T*x3)*UgA(x1 + x3) + x1*(0.0833333333333333*P*epsilon*x2**2 - 0.166666666666667*P*epsilon*x2*x3 + 0.0833333333333333*P*epsilon*x2 + 0.166666666666667*P*x2*x3**2 - 0.125*P*x2*x3 - 0.0416666666666667*P*x3**2 + 0.0416666666666667*P*x3 + 0.0833333333333333*R*epsilon*x3**2 - 0.0833333333333333*R*epsilon*x3 + 0.166666666666667*R*x2*x3**2 - 0.291666666666667*R*x2*x3 + 0.0833333333333333*R*x2 - 0.0833333333333333*R*x3**2 + 0.0833333333333333*R*x3 - 0.0833333333333333*S*epsilon*x3**2 + 0.0833333333333333*S*epsilon*x3 - 0.166666666666667*S*x2*x3**2 + 0.291666666666667*S*x2*x3 - 0.0833333333333333*S*x2 + 0.0833333333333333*S*x3**2 - 0.0833333333333333*S*x3 - 0.0833333333333333*T*epsilon*x2**2 + 0.166666666666667*T*epsilon*x2*x3 - 0.0833333333333333*T*epsilon*x2 - 0.166666666666667*T*x2*x3**2 + 0.125*T*x2*x3 + 0.0416666666666667*T*x3**2 - 0.0416666666666667*T*x3)*UGA(x1 + x3)**2 + x1*(0.0833333333333333*M*epsilon*x2**2 - 0.166666666666667*M*epsilon*x2*x3 + 0.0833333333333333*M*epsilon*x2 - 0.0833333333333333*M*epsilon*x3**2 + 0.0833333333333333*M*epsilon*x3 + 0.166666666666667*M*x2*x3 - 0.0833333333333333*M*x2 + 0.0416666666666667*M*x3**2 - 0.0416666666666667*M*x3 - 0.166666666666667*P*epsilon*x2**2 + 0.333333333333333*P*epsilon*x2*x3 - 0.166666666666667*P*epsilon*x2 - 0.333333333333333*P*x2*x3**2 + 0.25*P*x2*x3 + 0.0833333333333333*P*x3**2 - 0.0833333333333333*P*x3 + 0.0833333333333333*S*epsilon*x3**2 - 0.0833333333333333*S*epsilon*x3 + 0.166666666666667*S*x2*x3**2 - 0.291666666666667*S*x2*x3 + 0.0833333333333333*S*x2 - 0.0833333333333333*S*x3**2 + 0.0833333333333333*S*x3 + 0.0833333333333333*T*epsilon*x2**2 - 0.166666666666667*T*epsilon*x2*x3 + 0.0833333333333333*T*epsilon*x2 + 0.166666666666667*T*x2*x3**2 - 0.125*T*x2*x3 - 0.0416666666666667*T*x3**2 + 0.0416666666666667*T*x3)*UGA(x1 + x3) + x1*(-0.0833333333333333*P*epsilon*x2**2 - 0.166666666666667*P*epsilon*x2*x3 + 0.0833333333333333*P*epsilon*x2 - 0.166666666666667*P*x2*x3**2 + 0.208333333333333*P*x2*x3 + 0.166666666666667*P*x3**2 - 0.166666666666667*P*x3 + 0.0833333333333333*R*epsilon*x3**2 - 0.0833333333333333*R*epsilon*x3 - 0.166666666666667*R*x2*x3**2 + 0.208333333333333*R*x2*x3 - 0.0416666666666667*R*x2 + 0.125*R*x3**2 - 0.166666666666667*R*x3 + 0.0416666666666667*R - 0.0833333333333333*S*epsilon*x3**2 + 0.0833333333333333*S*epsilon*x3 + 0.166666666666667*S*x2*x3**2 - 0.208333333333333*S*x2*x3 + 0.0416666666666667*S*x2 - 0.125*S*x3**2 + 0.166666666666667*S*x3 - 0.0416666666666667*S + 0.0833333333333333*T*epsilon*x2**2 + 0.166666666666667*T*epsilon*x2*x3 - 0.0833333333333333*T*epsilon*x2 + 0.166666666666667*T*x2*x3**2 - 0.208333333333333*T*x2*x3 - 0.166666666666667*T*x3**2 + 0.166666666666667*T*x3)*UgA(x1 + x3)**2 + (-0.0833333333333333*P*epsilon*x2**2*x3 - 0.0833333333333333*P*epsilon*x2*x3**2 + 0.0833333333333333*P*epsilon*x2*x3 + 0.0416666666666667*P*x2*x3**2 + 0.0833333333333333*T*epsilon*x2**2*x3 + 0.0833333333333333*T*epsilon*x2*x3**2 - 0.0833333333333333*T*epsilon*x2*x3 - 0.0416666666666667*T*x2*x3**2)*UgA(x1 + x3)**2 + (0.166666666666667*P*epsilon*x2**2*x3 + 0.166666666666667*P*epsilon*x2*x3**2 - 0.166666666666667*P*epsilon*x2*x3 - 0.0833333333333333*P*x2*x3**2 - 0.0833333333333333*T*epsilon*x2**2*x3 - 0.0833333333333333*T*epsilon*x2*x3**2 + 0.0833333333333333*T*epsilon*x2*x3 + 0.0416666666666667*T*x2*x3**2)*UgA(x1 + x3) + (0.0833333333333333*P*epsilon*x2**2*x3 - 0.0833333333333333*P*epsilon*x2*x3**2 + 0.0833333333333333*P*epsilon*x2*x3 - 0.0416666666666667*R*x2*x3**2 + 0.0416666666666667*R*x2*x3 + 0.0416666666666667*S*x2*x3**2 - 0.0416666666666667*S*x2*x3 - 0.0833333333333333*T*epsilon*x2**2*x3 + 0.0833333333333333*T*epsilon*x2*x3**2 - 0.0833333333333333*T*epsilon*x2*x3)*UGA(x1 + x3)**2 + (0.0833333333333333*M*epsilon*x2**2*x3 - 0.0833333333333333*M*epsilon*x2*x3**2 + 0.0833333333333333*M*epsilon*x2*x3 + 0.0416666666666667*M*x2*x3**2 - 0.0416666666666667*M*x2*x3 - 0.166666666666667*P*epsilon*x2**2*x3 + 0.166666666666667*P*epsilon*x2*x3**2 - 0.166666666666667*P*epsilon*x2*x3 - 0.0416666666666667*S*x2*x3**2 + 0.0416666666666667*S*x2*x3 + 0.0833333333333333*T*epsilon*x2**2*x3 - 0.0833333333333333*T*epsilon*x2*x3**2 + 0.0833333333333333*T*epsilon*x2*x3)*UGA(x1 + x3))/(N*(1.0*x1 + 1.0*x3)*(1.0*x1 + 1.0*x3 - 1.0))"
      ]
     },
     "execution_count": 108,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(f1, UGA(x1 + x3), UgA(x1 + x3), x1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 105,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "12.0*(-0.166666666666667*M*x1**4 - 0.166666666666667*M*x1**3*x2 - 0.333333333333333*M*x1**3*x3 + 0.208333333333333*M*x1**3 - 0.333333333333333*M*x1**2*x2*x3 + 0.125*M*x1**2*x2 - 0.166666666666667*M*x1**2*x3**2 + 0.25*M*x1**2*x3 - 0.0416666666666667*M*x1**2 - 0.166666666666667*M*x1*x2*x3**2 + 0.125*M*x1*x2*x3 + 0.0416666666666667*M*x1*x3**2 - 0.0416666666666667*M*x1*x3 + 0.125*P*x1**3 + 0.0416666666666667*P*x1**2*x2 + 0.25*P*x1**2*x3 - 0.125*P*x1**2 + 0.0833333333333333*P*x1*x2*x3 + 0.125*P*x1*x3**2 - 0.125*P*x1*x3 + 0.0416666666666667*P*x2*x3**2 + epsilon*(0.0833333333333333*M*x1**2*x2 - 0.0833333333333333*M*x1*x2**2 + 0.166666666666667*M*x1*x2*x3 - 0.0833333333333333*M*x1*x2 - 0.0833333333333333*M*x2**2*x3 + 0.0833333333333333*M*x2*x3**2 - 0.0833333333333333*M*x2*x3 - 0.166666666666667*P*x1**2*x2 - 0.333333333333333*P*x1*x2*x3 + 0.166666666666667*P*x1*x2 - 0.166666666666667*P*x2*x3**2 + 0.166666666666667*P*x2*x3) + epsilon*(0.166666666666667*P*x1**2*x2 + 0.166666666666667*P*x1*x2**2 + 0.333333333333333*P*x1*x2*x3 - 0.166666666666667*P*x1*x2 + 0.166666666666667*P*x2**2*x3 + 0.166666666666667*P*x2*x3**2 - 0.166666666666667*P*x2*x3 + 0.0833333333333333*S*x1**2*x3 + 0.0833333333333333*S*x1*x3**2 - 0.0833333333333333*S*x1*x3 - 0.0833333333333333*T*x1**2*x2 - 0.0833333333333333*T*x1*x2**2 - 0.166666666666667*T*x1*x2*x3 + 0.0833333333333333*T*x1*x2 - 0.0833333333333333*T*x2**2*x3 - 0.0833333333333333*T*x2*x3**2 + 0.0833333333333333*T*x2*x3)*UgA(x1 + x3) + epsilon*(-0.0833333333333333*P*x1**2*x2 - 0.0833333333333333*P*x1*x2**2 - 0.166666666666667*P*x1*x2*x3 + 0.0833333333333333*P*x1*x2 - 0.0833333333333333*P*x2**2*x3 - 0.0833333333333333*P*x2*x3**2 + 0.0833333333333333*P*x2*x3 + 0.0833333333333333*R*x1**2*x3 + 0.0833333333333333*R*x1*x3**2 - 0.0833333333333333*R*x1*x3 - 0.0833333333333333*S*x1**2*x3 - 0.0833333333333333*S*x1*x3**2 + 0.0833333333333333*S*x1*x3 + 0.0833333333333333*T*x1**2*x2 + 0.0833333333333333*T*x1*x2**2 + 0.166666666666667*T*x1*x2*x3 - 0.0833333333333333*T*x1*x2 + 0.0833333333333333*T*x2**2*x3 + 0.0833333333333333*T*x2*x3**2 - 0.0833333333333333*T*x2*x3)*UgA(x1 + x3)**2 + epsilon*(-0.0833333333333333*P*x1**2*x2 + 0.0833333333333333*P*x1*x2**2 - 0.166666666666667*P*x1*x2*x3 + 0.0833333333333333*P*x1*x2 + 0.0833333333333333*P*x2**2*x3 - 0.0833333333333333*P*x2*x3**2 + 0.0833333333333333*P*x2*x3 + 0.166666666666667*R*x1**3 + 0.25*R*x1**2*x3 - 0.166666666666667*R*x1**2 + 0.0833333333333333*R*x1*x3**2 - 0.0833333333333333*R*x1*x3 - 0.166666666666667*S*x1**3 - 0.25*S*x1**2*x3 + 0.166666666666667*S*x1**2 - 0.0833333333333333*S*x1*x3**2 + 0.0833333333333333*S*x1*x3 + 0.0833333333333333*T*x1**2*x2 - 0.0833333333333333*T*x1*x2**2 + 0.166666666666667*T*x1*x2*x3 - 0.0833333333333333*T*x1*x2 - 0.0833333333333333*T*x2**2*x3 + 0.0833333333333333*T*x2*x3**2 - 0.0833333333333333*T*x2*x3)*UGA(x1 + x3)**2 + epsilon*(-0.166666666666667*M*x1**3 - 0.0833333333333333*M*x1**2*x2 - 0.25*M*x1**2*x3 + 0.166666666666667*M*x1**2 + 0.0833333333333333*M*x1*x2**2 - 0.166666666666667*M*x1*x2*x3 + 0.0833333333333333*M*x1*x2 - 0.0833333333333333*M*x1*x3**2 + 0.0833333333333333*M*x1*x3 + 0.0833333333333333*M*x2**2*x3 - 0.0833333333333333*M*x2*x3**2 + 0.0833333333333333*M*x2*x3 + 0.166666666666667*P*x1**2*x2 - 0.166666666666667*P*x1*x2**2 + 0.333333333333333*P*x1*x2*x3 - 0.166666666666667*P*x1*x2 - 0.166666666666667*P*x2**2*x3 + 0.166666666666667*P*x2*x3**2 - 0.166666666666667*P*x2*x3 + 0.166666666666667*S*x1**3 + 0.25*S*x1**2*x3 - 0.166666666666667*S*x1**2 + 0.0833333333333333*S*x1*x3**2 - 0.0833333333333333*S*x1*x3 - 0.0833333333333333*T*x1**2*x2 + 0.0833333333333333*T*x1*x2**2 - 0.166666666666667*T*x1*x2*x3 + 0.0833333333333333*T*x1*x2 + 0.0833333333333333*T*x2**2*x3 - 0.0833333333333333*T*x2*x3**2 + 0.0833333333333333*T*x2*x3)*UGA(x1 + x3) + (0.333333333333333*P*x1**4 + 0.333333333333333*P*x1**3*x2 + 0.666666666666667*P*x1**3*x3 - 0.666666666666667*P*x1**3 + 0.666666666666667*P*x1**2*x2*x3 - 0.333333333333333*P*x1**2*x2 + 0.333333333333333*P*x1**2*x3**2 - 1.0*P*x1**2*x3 + 0.333333333333333*P*x1**2 + 0.333333333333333*P*x1*x2*x3**2 - 0.416666666666667*P*x1*x2*x3 - 0.333333333333333*P*x1*x3**2 + 0.333333333333333*P*x1*x3 - 0.0833333333333333*P*x2*x3**2 - 0.166666666666667*S*x1**4 - 0.166666666666667*S*x1**3*x2 - 0.333333333333333*S*x1**3*x3 + 0.375*S*x1**3 - 0.333333333333333*S*x1**2*x2*x3 + 0.208333333333333*S*x1**2*x2 - 0.166666666666667*S*x1**2*x3**2 + 0.5*S*x1**2*x3 - 0.25*S*x1**2 - 0.166666666666667*S*x1*x2*x3**2 + 0.208333333333333*S*x1*x2*x3 - 0.0416666666666667*S*x1*x2 + 0.125*S*x1*x3**2 - 0.166666666666667*S*x1*x3 + 0.0416666666666667*S*x1 - 0.166666666666667*T*x1**4 - 0.166666666666667*T*x1**3*x2 - 0.333333333333333*T*x1**3*x3 + 0.333333333333333*T*x1**3 - 0.333333333333333*T*x1**2*x2*x3 + 0.166666666666667*T*x1**2*x2 - 0.166666666666667*T*x1**2*x3**2 + 0.5*T*x1**2*x3 - 0.166666666666667*T*x1**2 - 0.166666666666667*T*x1*x2*x3**2 + 0.208333333333333*T*x1*x2*x3 + 0.166666666666667*T*x1*x3**2 - 0.166666666666667*T*x1*x3 + 0.0416666666666667*T*x2*x3**2)*UgA(x1 + x3) + (0.125*M*x1**3 + 0.125*M*x1**2*x2 + 0.166666666666667*M*x1**2*x3 - 0.125*M*x1**2 + 0.166666666666667*M*x1*x2*x3 - 0.0833333333333333*M*x1*x2 + 0.0416666666666667*M*x1*x3**2 - 0.0416666666666667*M*x1*x3 + 0.0416666666666667*M*x2*x3**2 - 0.0416666666666667*M*x2*x3 - 0.333333333333333*P*x1**4 - 0.333333333333333*P*x1**3*x2 - 0.666666666666667*P*x1**3*x3 + 0.416666666666667*P*x1**3 - 0.666666666666667*P*x1**2*x2*x3 + 0.25*P*x1**2*x2 - 0.333333333333333*P*x1**2*x3**2 + 0.5*P*x1**2*x3 - 0.0833333333333333*P*x1**2 - 0.333333333333333*P*x1*x2*x3**2 + 0.25*P*x1*x2*x3 + 0.0833333333333333*P*x1*x3**2 - 0.0833333333333333*P*x1*x3 + 0.166666666666667*S*x1**4 + 0.166666666666667*S*x1**3*x2 + 0.333333333333333*S*x1**3*x3 - 0.333333333333333*S*x1**3 + 0.333333333333333*S*x1**2*x2*x3 - 0.25*S*x1**2*x2 + 0.166666666666667*S*x1**2*x3**2 - 0.416666666666667*S*x1**2*x3 + 0.166666666666667*S*x1**2 + 0.166666666666667*S*x1*x2*x3**2 - 0.291666666666667*S*x1*x2*x3 + 0.0833333333333333*S*x1*x2 - 0.0833333333333333*S*x1*x3**2 + 0.0833333333333333*S*x1*x3 - 0.0416666666666667*S*x2*x3**2 + 0.0416666666666667*S*x2*x3 + 0.166666666666667*T*x1**4 + 0.166666666666667*T*x1**3*x2 + 0.333333333333333*T*x1**3*x3 - 0.208333333333333*T*x1**3 + 0.333333333333333*T*x1**2*x2*x3 - 0.125*T*x1**2*x2 + 0.166666666666667*T*x1**2*x3**2 - 0.25*T*x1**2*x3 + 0.0416666666666667*T*x1**2 + 0.166666666666667*T*x1*x2*x3**2 - 0.125*T*x1*x2*x3 - 0.0416666666666667*T*x1*x3**2 + 0.0416666666666667*T*x1*x3)*UGA(x1 + x3) + (-0.166666666666667*P*x1**4 - 0.166666666666667*P*x1**3*x2 - 0.333333333333333*P*x1**3*x3 + 0.333333333333333*P*x1**3 - 0.333333333333333*P*x1**2*x2*x3 + 0.166666666666667*P*x1**2*x2 - 0.166666666666667*P*x1**2*x3**2 + 0.5*P*x1**2*x3 - 0.166666666666667*P*x1**2 - 0.166666666666667*P*x1*x2*x3**2 + 0.208333333333333*P*x1*x2*x3 + 0.166666666666667*P*x1*x3**2 - 0.166666666666667*P*x1*x3 + 0.0416666666666667*P*x2*x3**2 - 0.166666666666667*R*x1**4 - 0.166666666666667*R*x1**3*x2 - 0.333333333333333*R*x1**3*x3 + 0.375*R*x1**3 - 0.333333333333333*R*x1**2*x2*x3 + 0.208333333333333*R*x1**2*x2 - 0.166666666666667*R*x1**2*x3**2 + 0.5*R*x1**2*x3 - 0.25*R*x1**2 - 0.166666666666667*R*x1*x2*x3**2 + 0.208333333333333*R*x1*x2*x3 - 0.0416666666666667*R*x1*x2 + 0.125*R*x1*x3**2 - 0.166666666666667*R*x1*x3 + 0.0416666666666667*R*x1 + 0.166666666666667*S*x1**4 + 0.166666666666667*S*x1**3*x2 + 0.333333333333333*S*x1**3*x3 - 0.375*S*x1**3 + 0.333333333333333*S*x1**2*x2*x3 - 0.208333333333333*S*x1**2*x2 + 0.166666666666667*S*x1**2*x3**2 - 0.5*S*x1**2*x3 + 0.25*S*x1**2 + 0.166666666666667*S*x1*x2*x3**2 - 0.208333333333333*S*x1*x2*x3 + 0.0416666666666667*S*x1*x2 - 0.125*S*x1*x3**2 + 0.166666666666667*S*x1*x3 - 0.0416666666666667*S*x1 + 0.166666666666667*T*x1**4 + 0.166666666666667*T*x1**3*x2 + 0.333333333333333*T*x1**3*x3 - 0.333333333333333*T*x1**3 + 0.333333333333333*T*x1**2*x2*x3 - 0.166666666666667*T*x1**2*x2 + 0.166666666666667*T*x1**2*x3**2 - 0.5*T*x1**2*x3 + 0.166666666666667*T*x1**2 + 0.166666666666667*T*x1*x2*x3**2 - 0.208333333333333*T*x1*x2*x3 - 0.166666666666667*T*x1*x3**2 + 0.166666666666667*T*x1*x3 - 0.0416666666666667*T*x2*x3**2)*UgA(x1 + x3)**2 + (0.166666666666667*P*x1**4 + 0.166666666666667*P*x1**3*x2 + 0.333333333333333*P*x1**3*x3 - 0.208333333333333*P*x1**3 + 0.333333333333333*P*x1**2*x2*x3 - 0.125*P*x1**2*x2 + 0.166666666666667*P*x1**2*x3**2 - 0.25*P*x1**2*x3 + 0.0416666666666667*P*x1**2 + 0.166666666666667*P*x1*x2*x3**2 - 0.125*P*x1*x2*x3 - 0.0416666666666667*P*x1*x3**2 + 0.0416666666666667*P*x1*x3 + 0.166666666666667*R*x1**4 + 0.166666666666667*R*x1**3*x2 + 0.333333333333333*R*x1**3*x3 - 0.333333333333333*R*x1**3 + 0.333333333333333*R*x1**2*x2*x3 - 0.25*R*x1**2*x2 + 0.166666666666667*R*x1**2*x3**2 - 0.416666666666667*R*x1**2*x3 + 0.166666666666667*R*x1**2 + 0.166666666666667*R*x1*x2*x3**2 - 0.291666666666667*R*x1*x2*x3 + 0.0833333333333333*R*x1*x2 - 0.0833333333333333*R*x1*x3**2 + 0.0833333333333333*R*x1*x3 - 0.0416666666666667*R*x2*x3**2 + 0.0416666666666667*R*x2*x3 - 0.166666666666667*S*x1**4 - 0.166666666666667*S*x1**3*x2 - 0.333333333333333*S*x1**3*x3 + 0.333333333333333*S*x1**3 - 0.333333333333333*S*x1**2*x2*x3 + 0.25*S*x1**2*x2 - 0.166666666666667*S*x1**2*x3**2 + 0.416666666666667*S*x1**2*x3 - 0.166666666666667*S*x1**2 - 0.166666666666667*S*x1*x2*x3**2 + 0.291666666666667*S*x1*x2*x3 - 0.0833333333333333*S*x1*x2 + 0.0833333333333333*S*x1*x3**2 - 0.0833333333333333*S*x1*x3 + 0.0416666666666667*S*x2*x3**2 - 0.0416666666666667*S*x2*x3 - 0.166666666666667*T*x1**4 - 0.166666666666667*T*x1**3*x2 - 0.333333333333333*T*x1**3*x3 + 0.208333333333333*T*x1**3 - 0.333333333333333*T*x1**2*x2*x3 + 0.125*T*x1**2*x2 - 0.166666666666667*T*x1**2*x3**2 + 0.25*T*x1**2*x3 - 0.0416666666666667*T*x1**2 - 0.166666666666667*T*x1*x2*x3**2 + 0.125*T*x1*x2*x3 + 0.0416666666666667*T*x1*x3**2 - 0.0416666666666667*T*x1*x3)*UGA(x1 + x3)**2)/(N*(-4.0*x1**2 - 8.0*x1*x3 + 4.0*x1 - 4.0*x3**2 + 4.0*x3))"
      ]
     },
     "execution_count": 105,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(f1, UGA(x1 + x3), UgA(x1 + x3), epsilon)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "9.0*(-0.111111111111111*M*epsilon*x1**2*(-x1 + 1) + 0.111111111111111*M*epsilon*x1*(-x1 + 1)**2 + 0.111111111111111*M*epsilon*x1*(-x1 + 1) - 0.222222222222222*M*x1**3*(-x1 + 1) + 0.0555555555555556*M*x1**3 - 0.222222222222222*M*x1**2*(-x1 + 1)**2 + 0.277777777777778*M*x1**2*(-x1 + 1) - 0.0555555555555556*M*x1**2 + 0.111111111111111*M*x1*(-x1 + 1)**2 - 0.111111111111111*M*x1*(-x1 + 1) + 0.222222222222222*P*epsilon*x1**2*(-x1 + 1) - 0.222222222222222*P*epsilon*x1*(-x1 + 1) - 0.0555555555555556*P*x1**3 + 0.0555555555555556*P*x1**2*(-x1 + 1) + 0.0555555555555556*P*x1**2 + (-0.222222222222222*P*epsilon*x1**2*(-x1 + 1) - 0.222222222222222*P*epsilon*x1*(-x1 + 1)**2 + 0.222222222222222*P*epsilon*x1*(-x1 + 1) + 0.444444444444444*P*x1**3*(-x1 + 1) + 0.444444444444444*P*x1**2*(-x1 + 1)**2 - 0.666666666666667*P*x1**2*(-x1 + 1) - 0.222222222222222*P*x1*(-x1 + 1)**2 + 0.222222222222222*P*x1*(-x1 + 1) - 0.222222222222222*S*x1**3*(-x1 + 1) + 0.0555555555555556*S*x1**3 - 0.222222222222222*S*x1**2*(-x1 + 1)**2 + 0.5*S*x1**2*(-x1 + 1) - 0.111111111111111*S*x1**2 + 0.222222222222222*S*x1*(-x1 + 1)**2 - 0.277777777777778*S*x1*(-x1 + 1) + 0.0555555555555556*S*x1 + 0.111111111111111*T*epsilon*x1**2*(-x1 + 1) + 0.111111111111111*T*epsilon*x1*(-x1 + 1)**2 - 0.111111111111111*T*epsilon*x1*(-x1 + 1) - 0.222222222222222*T*x1**3*(-x1 + 1) - 0.222222222222222*T*x1**2*(-x1 + 1)**2 + 0.333333333333333*T*x1**2*(-x1 + 1) + 0.111111111111111*T*x1*(-x1 + 1)**2 - 0.111111111111111*T*x1*(-x1 + 1))*UgA(x1) + (0.111111111111111*P*epsilon*x1**2*(-x1 + 1) + 0.111111111111111*P*epsilon*x1*(-x1 + 1)**2 - 0.111111111111111*P*epsilon*x1*(-x1 + 1) - 0.222222222222222*P*x1**3*(-x1 + 1) - 0.222222222222222*P*x1**2*(-x1 + 1)**2 + 0.333333333333333*P*x1**2*(-x1 + 1) + 0.111111111111111*P*x1*(-x1 + 1)**2 - 0.111111111111111*P*x1*(-x1 + 1) - 0.222222222222222*R*x1**3*(-x1 + 1) + 0.0555555555555556*R*x1**3 - 0.222222222222222*R*x1**2*(-x1 + 1)**2 + 0.5*R*x1**2*(-x1 + 1) - 0.111111111111111*R*x1**2 + 0.222222222222222*R*x1*(-x1 + 1)**2 - 0.277777777777778*R*x1*(-x1 + 1) + 0.0555555555555556*R*x1 + 0.222222222222222*S*x1**3*(-x1 + 1) - 0.0555555555555556*S*x1**3 + 0.222222222222222*S*x1**2*(-x1 + 1)**2 - 0.5*S*x1**2*(-x1 + 1) + 0.111111111111111*S*x1**2 - 0.222222222222222*S*x1*(-x1 + 1)**2 + 0.277777777777778*S*x1*(-x1 + 1) - 0.0555555555555556*S*x1 - 0.111111111111111*T*epsilon*x1**2*(-x1 + 1) - 0.111111111111111*T*epsilon*x1*(-x1 + 1)**2 + 0.111111111111111*T*epsilon*x1*(-x1 + 1) + 0.222222222222222*T*x1**3*(-x1 + 1) + 0.222222222222222*T*x1**2*(-x1 + 1)**2 - 0.333333333333333*T*x1**2*(-x1 + 1) - 0.111111111111111*T*x1*(-x1 + 1)**2 + 0.111111111111111*T*x1*(-x1 + 1))*UgA(x1)**2 + (0.111111111111111*P*epsilon*x1**2*(-x1 + 1) - 0.111111111111111*P*epsilon*x1*(-x1 + 1)**2 - 0.111111111111111*P*epsilon*x1*(-x1 + 1) + 0.222222222222222*P*x1**3*(-x1 + 1) - 0.0555555555555556*P*x1**3 + 0.222222222222222*P*x1**2*(-x1 + 1)**2 - 0.277777777777778*P*x1**2*(-x1 + 1) + 0.0555555555555556*P*x1**2 - 0.111111111111111*P*x1*(-x1 + 1)**2 + 0.111111111111111*P*x1*(-x1 + 1) - 0.222222222222222*R*epsilon*x1**3 + 0.222222222222222*R*epsilon*x1**2 + 0.222222222222222*R*x1**3*(-x1 + 1) + 0.222222222222222*R*x1**2*(-x1 + 1)**2 - 0.333333333333333*R*x1**2*(-x1 + 1) - 0.222222222222222*R*x1*(-x1 + 1)**2 + 0.111111111111111*R*x1*(-x1 + 1) + 0.222222222222222*S*epsilon*x1**3 - 0.222222222222222*S*epsilon*x1**2 - 0.222222222222222*S*x1**3*(-x1 + 1) - 0.222222222222222*S*x1**2*(-x1 + 1)**2 + 0.333333333333333*S*x1**2*(-x1 + 1) + 0.222222222222222*S*x1*(-x1 + 1)**2 - 0.111111111111111*S*x1*(-x1 + 1) - 0.111111111111111*T*epsilon*x1**2*(-x1 + 1) + 0.111111111111111*T*epsilon*x1*(-x1 + 1)**2 + 0.111111111111111*T*epsilon*x1*(-x1 + 1) - 0.222222222222222*T*x1**3*(-x1 + 1) + 0.0555555555555556*T*x1**3 - 0.222222222222222*T*x1**2*(-x1 + 1)**2 + 0.277777777777778*T*x1**2*(-x1 + 1) - 0.0555555555555556*T*x1**2 + 0.111111111111111*T*x1*(-x1 + 1)**2 - 0.111111111111111*T*x1*(-x1 + 1))*UGA(x1)**2 + (0.222222222222222*M*epsilon*x1**3 + 0.111111111111111*M*epsilon*x1**2*(-x1 + 1) - 0.222222222222222*M*epsilon*x1**2 - 0.111111111111111*M*epsilon*x1*(-x1 + 1)**2 - 0.111111111111111*M*epsilon*x1*(-x1 + 1) - 0.0555555555555556*M*x1**3 + 0.0555555555555556*M*x1**2*(-x1 + 1) + 0.0555555555555556*M*x1**2 + 0.111111111111111*M*x1*(-x1 + 1)**2 - 0.222222222222222*P*epsilon*x1**2*(-x1 + 1) + 0.222222222222222*P*epsilon*x1*(-x1 + 1)**2 + 0.222222222222222*P*epsilon*x1*(-x1 + 1) - 0.444444444444444*P*x1**3*(-x1 + 1) + 0.111111111111111*P*x1**3 - 0.444444444444444*P*x1**2*(-x1 + 1)**2 + 0.555555555555556*P*x1**2*(-x1 + 1) - 0.111111111111111*P*x1**2 + 0.222222222222222*P*x1*(-x1 + 1)**2 - 0.222222222222222*P*x1*(-x1 + 1) - 0.222222222222222*S*epsilon*x1**3 + 0.222222222222222*S*epsilon*x1**2 + 0.222222222222222*S*x1**3*(-x1 + 1) + 0.222222222222222*S*x1**2*(-x1 + 1)**2 - 0.333333333333333*S*x1**2*(-x1 + 1) - 0.222222222222222*S*x1*(-x1 + 1)**2 + 0.111111111111111*S*x1*(-x1 + 1) + 0.111111111111111*T*epsilon*x1**2*(-x1 + 1) - 0.111111111111111*T*epsilon*x1*(-x1 + 1)**2 - 0.111111111111111*T*epsilon*x1*(-x1 + 1) + 0.222222222222222*T*x1**3*(-x1 + 1) - 0.0555555555555556*T*x1**3 + 0.222222222222222*T*x1**2*(-x1 + 1)**2 - 0.277777777777778*T*x1**2*(-x1 + 1) + 0.0555555555555556*T*x1**2 - 0.111111111111111*T*x1*(-x1 + 1)**2 + 0.111111111111111*T*x1*(-x1 + 1))*UGA(x1))/(N*(-4.0*x1**2 + 4.0*x1))"
      ]
     },
     "execution_count": 134,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f2.subs({x2: 1 - x1, x3:0 })"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "F = sym.Matrix([f1, f2, f3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [],
   "source": [
    "F_jac = F.jacobian([x1, x2, x3])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [],
   "source": [
    "UGA_star, UgA_star = sym.symbols(\"UGA_star, UgA_star\", nonnegative=True, real=True)\n",
    "UGA_prime_star = sym.symbols(\"UGA_prime_star\", real=True)\n",
    "\n",
    "evaluated_F_jac = (F_jac.subs({x2: 1 - x1, x3: 0})\n",
    "                        .subs({UGA(x1): UGA_star, UgA(x1): UgA_star})\n",
    "                        .doit()\n",
    "                        .subs({sym.Derivative(UGA(x1), x1): UGA_prime_star})\n",
    "                        .subs({x1: x1_star.subs({U_star: UGA_star})}))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [],
   "source": [
    "N_G = sym.symbols(\"N_G\", nonnegative=True, real=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4.0*(-0.5*M*epsilon + 0.5*P*epsilon + UGA_star**2*(0.5*P*epsilon + 0.25*R - 0.25*S - 0.5*T*epsilon) + UGA_star*(0.5*M*epsilon - 0.25*M - 1.0*P*epsilon + 0.25*S + 0.5*T*epsilon))"
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(2.0*M*UGA_star*epsilon - M*UGA_star - 2.0*M*epsilon + 2.0*P*UGA_star**2*epsilon - 4.0*P*UGA_star*epsilon + 2.0*P*epsilon + R*UGA_star**2 - S*UGA_star**2 + S*UGA_star - 2.0*T*UGA_star**2*epsilon + 2.0*T*UGA_star*epsilon, UGA_star)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 63,
   "metadata": {},
   "outputs": [],
   "source": [
    "n, d = sym.fraction(sym.together(evaluated_F_jac[0,0].subs({(-M + P*UGA_star**2 - 2.0*P*UGA_star + P + R*UGA_star**2 - S*UGA_star**2 + S*UGA_star - T*UGA_star**2 + T*UGA_star): 0.5 * N_G})))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 72,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "N_G*(-20736.0*epsilon - 10368.0)*(0.00617283950617284*M**4*epsilon**3 - 0.00308641975308642*M**4*epsilon**2 - 0.0185185185185185*M**3*P*epsilon**3 + 0.00925925925925926*M**3*P*epsilon**2 + 0.0185185185185185*M**2*P**2*epsilon**3 - 0.00925925925925926*M**2*P**2*epsilon**2 - 0.00617283950617284*M*P**3*epsilon**3 + 0.00308641975308642*M*P**3*epsilon**2 + N_G**2*UGA_star**4*(0.000771604938271605*P**2*epsilon**3 + 0.00154320987654321*P**2*epsilon**2 + 0.000964506172839506*P**2*epsilon + 0.000192901234567901*P**2 + 0.00231481481481481*P*R*epsilon**3 + 0.00385802469135802*P*R*epsilon**2 + 0.00212191358024691*P*R*epsilon + 0.000385802469135802*P*R - 0.00231481481481481*P*S*epsilon**3 - 0.00385802469135802*P*S*epsilon**2 - 0.00212191358024691*P*S*epsilon - 0.000385802469135802*P*S - 0.00154320987654321*P*T*epsilon**3 - 0.00308641975308642*P*T*epsilon**2 - 0.00192901234567901*P*T*epsilon - 0.000385802469135802*P*T + 0.00154320987654321*R**2*epsilon**3 + 0.00231481481481481*R**2*epsilon**2 + 0.00115740740740741*R**2*epsilon + 0.000192901234567901*R**2 - 0.00308641975308642*R*S*epsilon**3 - 0.00462962962962963*R*S*epsilon**2 - 0.00231481481481481*R*S*epsilon - 0.000385802469135802*R*S - 0.00231481481481481*R*T*epsilon**3 - 0.00385802469135802*R*T*epsilon**2 - 0.00212191358024691*R*T*epsilon - 0.000385802469135802*R*T + 0.00154320987654321*S**2*epsilon**3 + 0.00231481481481481*S**2*epsilon**2 + 0.00115740740740741*S**2*epsilon + 0.000192901234567901*S**2 + 0.00231481481481481*S*T*epsilon**3 + 0.00385802469135802*S*T*epsilon**2 + 0.00212191358024691*S*T*epsilon + 0.000385802469135802*S*T + 0.000771604938271605*T**2*epsilon**3 + 0.00154320987654321*T**2*epsilon**2 + 0.000964506172839506*T**2*epsilon + 0.000192901234567901*T**2) + N_G**2*UGA_star**3*(-0.000771604938271605*M*P*epsilon**3 - 0.000771604938271605*M*P*epsilon**2 - 0.000192901234567901*M*P*epsilon - 0.000771604938271605*M*R*epsilon**3 - 0.000771604938271605*M*R*epsilon**2 - 0.000192901234567901*M*R*epsilon + 0.000771604938271605*M*S*epsilon**3 + 0.000771604938271605*M*S*epsilon**2 + 0.000192901234567901*M*S*epsilon + 0.000771604938271605*M*T*epsilon**3 + 0.000771604938271605*M*T*epsilon**2 + 0.000192901234567901*M*T*epsilon - 0.00308641975308642*P**2*epsilon**3 - 0.00617283950617284*P**2*epsilon**2 - 0.00385802469135802*P**2*epsilon - 0.000771604938271605*P**2 - 0.00462962962962963*P*R*epsilon**3 - 0.00771604938271605*P*R*epsilon**2 - 0.00424382716049383*P*R*epsilon - 0.000771604938271605*P*R + 0.00694444444444444*P*S*epsilon**3 + 0.0115740740740741*P*S*epsilon**2 + 0.00636574074074074*P*S*epsilon + 0.00115740740740741*P*S + 0.00462962962962963*P*T*epsilon**3 + 0.00925925925925926*P*T*epsilon**2 + 0.00578703703703704*P*T*epsilon + 0.00115740740740741*P*T + 0.00308641975308642*R*S*epsilon**3 + 0.00462962962962963*R*S*epsilon**2 + 0.00231481481481481*R*S*epsilon + 0.000385802469135802*R*S + 0.00231481481481481*R*T*epsilon**3 + 0.00385802469135802*R*T*epsilon**2 + 0.00212191358024691*R*T*epsilon + 0.000385802469135802*R*T - 0.00308641975308642*S**2*epsilon**3 - 0.00462962962962963*S**2*epsilon**2 - 0.00231481481481481*S**2*epsilon - 0.000385802469135802*S**2 - 0.00462962962962963*S*T*epsilon**3 - 0.00771604938271605*S*T*epsilon**2 - 0.00424382716049383*S*T*epsilon - 0.000771604938271605*S*T - 0.00154320987654321*T**2*epsilon**3 - 0.00308641975308642*T**2*epsilon**2 - 0.00192901234567901*T**2*epsilon - 0.000385802469135802*T**2) + N_G**2*UGA_star**2*UgA_star**2*(-0.000771604938271605*P**2*epsilon**3 - 0.000771604938271605*P**2*epsilon**2 - 0.000192901234567901*P**2*epsilon - 0.000771604938271605*P*R*epsilon**3 - 0.00115740740740741*P*R*epsilon**2 - 0.000578703703703704*P*R*epsilon - 9.64506172839506e-5*P*R + 0.000771604938271605*P*S*epsilon**3 + 0.00115740740740741*P*S*epsilon**2 + 0.000578703703703704*P*S*epsilon + 9.64506172839506e-5*P*S + 0.00154320987654321*P*T*epsilon**3 + 0.00154320987654321*P*T*epsilon**2 + 0.000385802469135802*P*T*epsilon - 0.000385802469135802*R**2*epsilon**2 - 0.000385802469135802*R**2*epsilon - 9.64506172839506e-5*R**2 + 0.000771604938271605*R*S*epsilon**2 + 0.000771604938271605*R*S*epsilon + 0.000192901234567901*R*S + 0.000771604938271605*R*T*epsilon**3 + 0.00115740740740741*R*T*epsilon**2 + 0.000578703703703704*R*T*epsilon + 9.64506172839506e-5*R*T - 0.000385802469135802*S**2*epsilon**2 - 0.000385802469135802*S**2*epsilon - 9.64506172839506e-5*S**2 - 0.000771604938271605*S*T*epsilon**3 - 0.00115740740740741*S*T*epsilon**2 - 0.000578703703703704*S*T*epsilon - 9.64506172839506e-5*S*T - 0.000771604938271605*T**2*epsilon**3 - 0.000771604938271605*T**2*epsilon**2 - 0.000192901234567901*T**2*epsilon) + N_G**2*UGA_star**2*UgA_star*(0.00154320987654321*P**2*epsilon**3 + 0.00154320987654321*P**2*epsilon**2 + 0.000385802469135802*P**2*epsilon - 0.00154320987654321*P*R*epsilon**3 + 0.00115740740740741*P*R*epsilon + 0.000385802469135802*P*R - 0.00115740740740741*P*S*epsilon**2 - 0.00115740740740741*P*S*epsilon - 0.000289351851851852*P*S - 0.00231481481481481*P*T*epsilon**3 - 0.00231481481481481*P*T*epsilon**2 - 0.000578703703703704*P*T*epsilon - 0.000385802469135802*R*S*epsilon**2 - 0.000385802469135802*R*S*epsilon - 9.64506172839506e-5*R*S + 0.000771604938271605*R*T*epsilon**3 - 0.000578703703703704*R*T*epsilon - 0.000192901234567901*R*T + 0.000385802469135802*S**2*epsilon**2 + 0.000385802469135802*S**2*epsilon + 9.64506172839506e-5*S**2 + 0.000771604938271605*S*T*epsilon**3 + 0.00115740740740741*S*T*epsilon**2 + 0.000578703703703704*S*T*epsilon + 9.64506172839506e-5*S*T + 0.000771604938271605*T**2*epsilon**3 + 0.000771604938271605*T**2*epsilon**2 + 0.000192901234567901*T**2*epsilon) + N_G**2*UGA_star**2*(0.00154320987654321*M*P*UGA_prime_star*epsilon**3 + 0.000771604938271605*M*P*UGA_prime_star*epsilon**2 - 0.000385802469135802*M*P*UGA_prime_star*epsilon - 0.000192901234567901*M*P*UGA_prime_star - 0.00154320987654321*M*P*epsilon**2 - 0.00154320987654321*M*P*epsilon - 0.000385802469135802*M*P + 0.00154320987654321*M*R*UGA_prime_star*epsilon**3 + 0.000771604938271605*M*R*UGA_prime_star*epsilon**2 - 0.000385802469135802*M*R*UGA_prime_star*epsilon - 0.000192901234567901*M*R*UGA_prime_star - 0.00231481481481481*M*R*epsilon**3 - 0.00385802469135802*M*R*epsilon**2 - 0.00212191358024691*M*R*epsilon - 0.000385802469135802*M*R - 0.00154320987654321*M*S*UGA_prime_star*epsilon**3 - 0.000771604938271605*M*S*UGA_prime_star*epsilon**2 + 0.000385802469135802*M*S*UGA_prime_star*epsilon + 0.000192901234567901*M*S*UGA_prime_star + 0.00154320987654321*M*S*epsilon**3 + 0.00308641975308642*M*S*epsilon**2 + 0.00192901234567901*M*S*epsilon + 0.000385802469135802*M*S - 0.00154320987654321*M*T*UGA_prime_star*epsilon**3 - 0.000771604938271605*M*T*UGA_prime_star*epsilon**2 + 0.000385802469135802*M*T*UGA_prime_star*epsilon + 0.000192901234567901*M*T*UGA_prime_star + 0.000771604938271605*M*T*epsilon**3 + 0.00231481481481481*M*T*epsilon**2 + 0.00173611111111111*M*T*epsilon + 0.000385802469135802*M*T + 0.00385802469135802*P**2*epsilon**3 + 0.00848765432098765*P**2*epsilon**2 + 0.00559413580246914*P**2*epsilon + 0.00115740740740741*P**2 - 0.00308641975308642*P*R*UGA_prime_star*epsilon**3 - 0.00154320987654321*P*R*UGA_prime_star*epsilon**2 + 0.000771604938271605*P*R*UGA_prime_star*epsilon + 0.000385802469135802*P*R*UGA_prime_star + 0.00308641975308642*P*R*epsilon**3 + 0.00385802469135802*P*R*epsilon**2 + 0.00154320987654321*P*R*epsilon + 0.000192901234567901*P*R + 0.00154320987654321*P*S*UGA_prime_star*epsilon**3 + 0.000771604938271605*P*S*UGA_prime_star*epsilon**2 - 0.000385802469135802*P*S*UGA_prime_star*epsilon - 0.000192901234567901*P*S*UGA_prime_star - 0.00771604938271605*P*S*epsilon**3 - 0.0115740740740741*P*S*epsilon**2 - 0.00578703703703704*P*S*epsilon - 0.000964506172839506*P*S - 0.00385802469135802*P*T*epsilon**3 - 0.00848765432098765*P*T*epsilon**2 - 0.00559413580246914*P*T*epsilon - 0.00115740740740741*P*T + 0.00154320987654321*R*T*UGA_prime_star*epsilon**3 + 0.000771604938271605*R*T*UGA_prime_star*epsilon**2 - 0.000385802469135802*R*T*UGA_prime_star*epsilon - 0.000192901234567901*R*T*UGA_prime_star + 0.00154320987654321*S**2*epsilon**3 + 0.00231481481481481*S**2*epsilon**2 + 0.00115740740740741*S**2*epsilon + 0.000192901234567901*S**2 + 0.00231481481481481*S*T*epsilon**3 + 0.00385802469135802*S*T*epsilon**2 + 0.00212191358024691*S*T*epsilon + 0.000385802469135802*S*T + 0.000771604938271605*T**2*epsilon**3 + 0.00154320987654321*T**2*epsilon**2 + 0.000964506172839506*T**2*epsilon + 0.000192901234567901*T**2) + N_G**2*UGA_star*UgA_star**2*(-0.00154320987654321*M*P*epsilon**3 - 0.000771604938271605*M*P*epsilon**2 + 0.000385802469135802*M*P*epsilon + 0.000192901234567901*M*P - 0.00154320987654321*M*R*epsilon**3 - 0.000771604938271605*M*R*epsilon**2 + 0.000385802469135802*M*R*epsilon + 0.000192901234567901*M*R + 0.00154320987654321*M*S*epsilon**3 + 0.000771604938271605*M*S*epsilon**2 - 0.000385802469135802*M*S*epsilon - 0.000192901234567901*M*S + 0.00154320987654321*M*T*epsilon**3 + 0.000771604938271605*M*T*epsilon**2 - 0.000385802469135802*M*T*epsilon - 0.000192901234567901*M*T + 0.00154320987654321*P**2*epsilon**3 + 0.00154320987654321*P**2*epsilon**2 + 0.000385802469135802*P**2*epsilon + 0.00308641975308642*P*R*epsilon**3 + 0.00231481481481481*P*R*epsilon**2 - 0.000192901234567901*P*R - 0.00231481481481481*P*S*epsilon**3 - 0.00231481481481481*P*S*epsilon**2 - 0.000578703703703704*P*S*epsilon - 0.00231481481481481*P*T*epsilon**3 - 0.00231481481481481*P*T*epsilon**2 - 0.000578703703703704*P*T*epsilon - 0.000385802469135802*R*S*epsilon**2 - 0.000385802469135802*R*S*epsilon - 9.64506172839506e-5*R*S - 0.00154320987654321*R*T*epsilon**3 - 0.00115740740740741*R*T*epsilon**2 + 9.64506172839506e-5*R*T + 0.000385802469135802*S**2*epsilon**2 + 0.000385802469135802*S**2*epsilon + 9.64506172839506e-5*S**2 + 0.000771604938271605*S*T*epsilon**3 + 0.00115740740740741*S*T*epsilon**2 + 0.000578703703703704*S*T*epsilon + 9.64506172839506e-5*S*T + 0.000771604938271605*T**2*epsilon**3 + 0.000771604938271605*T**2*epsilon**2 + 0.000192901234567901*T**2*epsilon) + N_G**2*UGA_star*UgA_star*(0.00308641975308642*M*P*epsilon**3 + 0.00154320987654321*M*P*epsilon**2 - 0.000771604938271605*M*P*epsilon - 0.000385802469135802*M*P - 0.00154320987654321*M*S*epsilon**3 - 0.000771604938271605*M*S*epsilon**2 + 0.000385802469135802*M*S*epsilon + 0.000192901234567901*M*S - 0.00154320987654321*M*T*epsilon**3 - 0.000771604938271605*M*T*epsilon**2 + 0.000385802469135802*M*T*epsilon + 0.000192901234567901*M*T - 0.00308641975308642*P**2*epsilon**3 - 0.00308641975308642*P**2*epsilon**2 - 0.000771604938271605*P**2*epsilon + 0.00154320987654321*P*S*epsilon**3 + 0.00231481481481481*P*S*epsilon**2 + 0.00115740740740741*P*S*epsilon + 0.000192901234567901*P*S + 0.00308641975308642*P*T*epsilon**3 + 0.00308641975308642*P*T*epsilon**2 + 0.000771604938271605*P*T*epsilon - 0.000385802469135802*S**2*epsilon**2 - 0.000385802469135802*S**2*epsilon - 9.64506172839506e-5*S**2 - 0.000771604938271605*S*T*epsilon**3 - 0.00115740740740741*S*T*epsilon**2 - 0.000578703703703704*S*T*epsilon - 9.64506172839506e-5*S*T - 0.000771604938271605*T**2*epsilon**3 - 0.000771604938271605*T**2*epsilon**2 - 0.000192901234567901*T**2*epsilon) + N_G**2*UGA_star*(0.000771604938271605*M**2*epsilon**3 + 0.000771604938271605*M**2*epsilon**2 + 0.000192901234567901*M**2*epsilon + 0.000771604938271605*M*P*epsilon**3 + 0.00462962962962963*M*P*epsilon**2 + 0.00405092592592593*M*P*epsilon + 0.000964506172839506*M*P - 0.00308641975308642*M*R*UGA_prime_star*epsilon**3 - 0.00154320987654321*M*R*UGA_prime_star*epsilon**2 + 0.000771604938271605*M*R*UGA_prime_star*epsilon + 0.000385802469135802*M*R*UGA_prime_star + 0.00308641975308642*M*S*UGA_prime_star*epsilon**3 + 0.00154320987654321*M*S*UGA_prime_star*epsilon**2 - 0.000771604938271605*M*S*UGA_prime_star*epsilon - 0.000385802469135802*M*S*UGA_prime_star - 0.00231481481481481*M*S*epsilon**3 - 0.00385802469135802*M*S*epsilon**2 - 0.00212191358024691*M*S*epsilon - 0.000385802469135802*M*S - 0.00154320987654321*M*T*epsilon**3 - 0.00308641975308642*M*T*epsilon**2 - 0.00192901234567901*M*T*epsilon - 0.000385802469135802*M*T - 0.00154320987654321*P**2*epsilon**3 - 0.00462962962962963*P**2*epsilon**2 - 0.00347222222222222*P**2*epsilon - 0.000771604938271605*P**2 + 0.00308641975308642*P*R*UGA_prime_star*epsilon**3 + 0.00154320987654321*P*R*UGA_prime_star*epsilon**2 - 0.000771604938271605*P*R*UGA_prime_star*epsilon - 0.000385802469135802*P*R*UGA_prime_star - 0.00308641975308642*P*S*UGA_prime_star*epsilon**3 - 0.00154320987654321*P*S*UGA_prime_star*epsilon**2 + 0.000771604938271605*P*S*UGA_prime_star*epsilon + 0.000385802469135802*P*S*UGA_prime_star + 0.00308641975308642*P*S*epsilon**3 + 0.00385802469135802*P*S*epsilon**2 + 0.00154320987654321*P*S*epsilon + 0.000192901234567901*P*S + 0.000771604938271605*P*T*epsilon**3 + 0.00231481481481481*P*T*epsilon**2 + 0.00173611111111111*P*T*epsilon + 0.000385802469135802*P*T) + N_G**2*UgA_star**2*(0.000771604938271605*M*P*epsilon**3 + 0.000771604938271605*M*P*epsilon**2 + 0.000192901234567901*M*P*epsilon + 0.00154320987654321*M*R*epsilon**3 + 0.00115740740740741*M*R*epsilon**2 - 9.64506172839506e-5*M*R - 0.00154320987654321*M*S*epsilon**3 - 0.00115740740740741*M*S*epsilon**2 + 9.64506172839506e-5*M*S - 0.000771604938271605*M*T*epsilon**3 - 0.000771604938271605*M*T*epsilon**2 - 0.000192901234567901*M*T*epsilon - 0.000771604938271605*P**2*epsilon**3 - 0.000771604938271605*P**2*epsilon**2 - 0.000192901234567901*P**2*epsilon - 0.00154320987654321*P*R*epsilon**3 - 0.00115740740740741*P*R*epsilon**2 + 9.64506172839506e-5*P*R + 0.00154320987654321*P*S*epsilon**3 + 0.00115740740740741*P*S*epsilon**2 - 9.64506172839506e-5*P*S + 0.000771604938271605*P*T*epsilon**3 + 0.000771604938271605*P*T*epsilon**2 + 0.000192901234567901*P*T*epsilon) + N_G**2*UgA_star*(-0.00154320987654321*M*P*epsilon**3 - 0.00154320987654321*M*P*epsilon**2 - 0.000385802469135802*M*P*epsilon + 0.00154320987654321*M*S*epsilon**3 + 0.00115740740740741*M*S*epsilon**2 - 9.64506172839506e-5*M*S + 0.000771604938271605*M*T*epsilon**3 + 0.000771604938271605*M*T*epsilon**2 + 0.000192901234567901*M*T*epsilon + 0.00154320987654321*P**2*epsilon**3 + 0.00154320987654321*P**2*epsilon**2 + 0.000385802469135802*P**2*epsilon - 0.00154320987654321*P*S*epsilon**3 - 0.00115740740740741*P*S*epsilon**2 + 9.64506172839506e-5*P*S - 0.000771604938271605*P*T*epsilon**3 - 0.000771604938271605*P*T*epsilon**2 - 0.000192901234567901*P*T*epsilon) + N_G**2*(0.00154320987654321*M**2*UGA_prime_star*epsilon**3 + 0.000771604938271605*M**2*UGA_prime_star*epsilon**2 - 0.000385802469135802*M**2*UGA_prime_star*epsilon - 0.000192901234567901*M**2*UGA_prime_star + 0.000771604938271605*M**2*epsilon**3 + 0.00154320987654321*M**2*epsilon**2 + 0.000964506172839506*M**2*epsilon + 0.000192901234567901*M**2 - 0.00154320987654321*M*P*UGA_prime_star*epsilon**3 - 0.000771604938271605*M*P*UGA_prime_star*epsilon**2 + 0.000385802469135802*M*P*UGA_prime_star*epsilon + 0.000192901234567901*M*P*UGA_prime_star - 0.000771604938271605*M*P*epsilon**3 - 0.00231481481481481*M*P*epsilon**2 - 0.00173611111111111*M*P*epsilon - 0.000385802469135802*M*P - 0.00154320987654321*M*S*UGA_prime_star*epsilon**3 - 0.000771604938271605*M*S*UGA_prime_star*epsilon**2 + 0.000385802469135802*M*S*UGA_prime_star*epsilon + 0.000192901234567901*M*S*UGA_prime_star + 0.000771604938271605*P**2*epsilon**2 + 0.000771604938271605*P**2*epsilon + 0.000192901234567901*P**2 + 0.00154320987654321*P*S*UGA_prime_star*epsilon**3 + 0.000771604938271605*P*S*UGA_prime_star*epsilon**2 - 0.000385802469135802*P*S*UGA_prime_star*epsilon - 0.000192901234567901*P*S*UGA_prime_star) + N_G*UGA_star**6*(-0.00462962962962963*P**3*epsilon**3 - 0.00308641975308642*P**3*epsilon**2 - 0.000385802469135802*P**3*epsilon - 0.0108024691358025*P**2*R*epsilon**3 - 0.00848765432098765*P**2*R*epsilon**2 - 0.00192901234567901*P**2*R*epsilon - 0.000192901234567901*P**2*R + 0.0108024691358025*P**2*S*epsilon**3 + 0.00848765432098765*P**2*S*epsilon**2 + 0.00192901234567901*P**2*S*epsilon + 0.000192901234567901*P**2*S + 0.0138888888888889*P**2*T*epsilon**3 + 0.00925925925925926*P**2*T*epsilon**2 + 0.00115740740740741*P**2*T*epsilon - 0.00617283950617284*P*R**2*epsilon**3 - 0.00848765432098765*P*R**2*epsilon**2 - 0.00308641975308642*P*R**2*epsilon - 0.000192901234567901*P*R**2 + 0.0123456790123457*P*R*S*epsilon**3 + 0.0169753086419753*P*R*S*epsilon**2 + 0.00617283950617284*P*R*S*epsilon + 0.000385802469135802*P*R*S + 0.0216049382716049*P*R*T*epsilon**3 + 0.0169753086419753*P*R*T*epsilon**2 + 0.00385802469135802*P*R*T*epsilon + 0.000385802469135802*P*R*T - 0.00617283950617284*P*S**2*epsilon**3 - 0.00848765432098765*P*S**2*epsilon**2 - 0.00308641975308642*P*S**2*epsilon - 0.000192901234567901*P*S**2 - 0.0216049382716049*P*S*T*epsilon**3 - 0.0169753086419753*P*S*T*epsilon**2 - 0.00385802469135802*P*S*T*epsilon - 0.000385802469135802*P*S*T - 0.0138888888888889*P*T**2*epsilon**3 - 0.00925925925925926*P*T**2*epsilon**2 - 0.00115740740740741*P*T**2*epsilon - 0.00308641975308642*R**3*epsilon**2 - 0.00154320987654321*R**3*epsilon + 0.00925925925925926*R**2*S*epsilon**2 + 0.00462962962962963*R**2*S*epsilon + 0.00617283950617284*R**2*T*epsilon**3 + 0.00848765432098765*R**2*T*epsilon**2 + 0.00308641975308642*R**2*T*epsilon + 0.000192901234567901*R**2*T - 0.00925925925925926*R*S**2*epsilon**2 - 0.00462962962962963*R*S**2*epsilon - 0.0123456790123457*R*S*T*epsilon**3 - 0.0169753086419753*R*S*T*epsilon**2 - 0.00617283950617284*R*S*T*epsilon - 0.000385802469135802*R*S*T - 0.0108024691358025*R*T**2*epsilon**3 - 0.00848765432098765*R*T**2*epsilon**2 - 0.00192901234567901*R*T**2*epsilon - 0.000192901234567901*R*T**2 + 0.00308641975308642*S**3*epsilon**2 + 0.00154320987654321*S**3*epsilon + 0.00617283950617284*S**2*T*epsilon**3 + 0.00848765432098765*S**2*T*epsilon**2 + 0.00308641975308642*S**2*T*epsilon + 0.000192901234567901*S**2*T + 0.0108024691358025*S*T**2*epsilon**3 + 0.00848765432098765*S*T**2*epsilon**2 + 0.00192901234567901*S*T**2*epsilon + 0.000192901234567901*S*T**2 + 0.00462962962962963*T**3*epsilon**3 + 0.00308641975308642*T**3*epsilon**2 + 0.000385802469135802*T**3*epsilon) + N_G*UGA_star**5*(-0.00308641975308642*M*P**2*epsilon**3 - 0.000771604938271605*M*P**2*epsilon**2 + 0.000771604938271605*M*P**2*epsilon + 0.000192901234567901*M*P**2 - 0.00925925925925926*M*P*R*epsilon**3 + 0.00231481481481481*M*P*R*epsilon + 0.00925925925925926*M*P*S*epsilon**3 - 0.00231481481481481*M*P*S*epsilon + 0.00617283950617284*M*P*T*epsilon**3 + 0.00154320987654321*M*P*T*epsilon**2 - 0.00154320987654321*M*P*T*epsilon - 0.000385802469135802*M*P*T - 0.00617283950617284*M*R**2*epsilon**3 + 0.000771604938271605*M*R**2*epsilon**2 + 0.00154320987654321*M*R**2*epsilon - 0.000192901234567901*M*R**2 + 0.0123456790123457*M*R*S*epsilon**3 - 0.00154320987654321*M*R*S*epsilon**2 - 0.00308641975308642*M*R*S*epsilon + 0.000385802469135802*M*R*S + 0.00925925925925926*M*R*T*epsilon**3 - 0.00231481481481481*M*R*T*epsilon - 0.00617283950617284*M*S**2*epsilon**3 + 0.000771604938271605*M*S**2*epsilon**2 + 0.00154320987654321*M*S**2*epsilon - 0.000192901234567901*M*S**2 - 0.00925925925925926*M*S*T*epsilon**3 + 0.00231481481481481*M*S*T*epsilon - 0.00308641975308642*M*T**2*epsilon**3 - 0.000771604938271605*M*T**2*epsilon**2 + 0.000771604938271605*M*T**2*epsilon + 0.000192901234567901*M*T**2 + 0.0277777777777778*P**3*epsilon**3 + 0.0185185185185185*P**3*epsilon**2 + 0.00231481481481481*P**3*epsilon + 0.0432098765432099*P**2*R*epsilon**3 + 0.0339506172839506*P**2*R*epsilon**2 + 0.00771604938271605*P**2*R*epsilon + 0.000771604938271605*P**2*R - 0.0540123456790123*P**2*S*epsilon**3 - 0.0424382716049383*P**2*S*epsilon**2 - 0.00964506172839506*P**2*S*epsilon - 0.000964506172839506*P**2*S - 0.0694444444444444*P**2*T*epsilon**3 - 0.0462962962962963*P**2*T*epsilon**2 - 0.00578703703703704*P**2*T*epsilon + 0.0123456790123457*P*R**2*epsilon**3 + 0.0169753086419753*P*R**2*epsilon**2 + 0.00617283950617284*P*R**2*epsilon + 0.000385802469135802*P*R**2 - 0.037037037037037*P*R*S*epsilon**3 - 0.0509259259259259*P*R*S*epsilon**2 - 0.0185185185185185*P*R*S*epsilon - 0.00115740740740741*P*R*S - 0.0648148148148148*P*R*T*epsilon**3 - 0.0509259259259259*P*R*T*epsilon**2 - 0.0115740740740741*P*R*T*epsilon - 0.00115740740740741*P*R*T + 0.0246913580246914*P*S**2*epsilon**3 + 0.0339506172839506*P*S**2*epsilon**2 + 0.0123456790123457*P*S**2*epsilon + 0.000771604938271605*P*S**2 + 0.0864197530864197*P*S*T*epsilon**3 + 0.0679012345679012*P*S*T*epsilon**2 + 0.0154320987654321*P*S*T*epsilon + 0.00154320987654321*P*S*T + 0.0555555555555556*P*T**2*epsilon**3 + 0.037037037037037*P*T**2*epsilon**2 + 0.00462962962962963*P*T**2*epsilon - 0.00925925925925926*R**2*S*epsilon**2 - 0.00462962962962963*R**2*S*epsilon - 0.00617283950617284*R**2*T*epsilon**3 - 0.00848765432098765*R**2*T*epsilon**2 - 0.00308641975308642*R**2*T*epsilon - 0.000192901234567901*R**2*T + 0.0185185185185185*R*S**2*epsilon**2 + 0.00925925925925926*R*S**2*epsilon + 0.0246913580246914*R*S*T*epsilon**3 + 0.0339506172839506*R*S*T*epsilon**2 + 0.0123456790123457*R*S*T*epsilon + 0.000771604938271605*R*S*T + 0.0216049382716049*R*T**2*epsilon**3 + 0.0169753086419753*R*T**2*epsilon**2 + 0.00385802469135802*R*T**2*epsilon + 0.000385802469135802*R*T**2 - 0.00925925925925926*S**3*epsilon**2 - 0.00462962962962963*S**3*epsilon - 0.0185185185185185*S**2*T*epsilon**3 - 0.025462962962963*S**2*T*epsilon**2 - 0.00925925925925926*S**2*T*epsilon - 0.000578703703703704*S**2*T - 0.0324074074074074*S*T**2*epsilon**3 - 0.025462962962963*S*T**2*epsilon**2 - 0.00578703703703704*S*T**2*epsilon - 0.000578703703703704*S*T**2 - 0.0138888888888889*T**3*epsilon**3 - 0.00925925925925926*T**3*epsilon**2 - 0.00115740740740741*T**3*epsilon) + N_G*UGA_star**4*UgA_star**2*(0.00462962962962963*P**3*epsilon**3 + 0.000771604938271605*P**3*epsilon**2 - 0.000771604938271605*P**3*epsilon + 0.00771604938271605*P**2*R*epsilon**3 + 0.00385802469135802*P**2*R*epsilon**2 - 0.000771604938271605*P**2*R*epsilon - 0.000385802469135802*P**2*R - 0.00771604938271605*P**2*S*epsilon**3 - 0.00385802469135802*P**2*S*epsilon**2 + 0.000771604938271605*P**2*S*epsilon + 0.000385802469135802*P**2*S - 0.0138888888888889*P**2*T*epsilon**3 - 0.00231481481481481*P**2*T*epsilon**2 + 0.00231481481481481*P**2*T*epsilon + 0.00308641975308642*P*R**2*epsilon**3 + 0.00462962962962963*P*R**2*epsilon**2 + 0.000385802469135802*P*R**2*epsilon - 0.000578703703703704*P*R**2 - 0.00617283950617284*P*R*S*epsilon**3 - 0.00925925925925926*P*R*S*epsilon**2 - 0.000771604938271605*P*R*S*epsilon + 0.00115740740740741*P*R*S - 0.0154320987654321*P*R*T*epsilon**3 - 0.00771604938271605*P*R*T*epsilon**2 + 0.00154320987654321*P*R*T*epsilon + 0.000771604938271605*P*R*T + 0.00308641975308642*P*S**2*epsilon**3 + 0.00462962962962963*P*S**2*epsilon**2 + 0.000385802469135802*P*S**2*epsilon - 0.000578703703703704*P*S**2 + 0.0154320987654321*P*S*T*epsilon**3 + 0.00771604938271605*P*S*T*epsilon**2 - 0.00154320987654321*P*S*T*epsilon - 0.000771604938271605*P*S*T + 0.0138888888888889*P*T**2*epsilon**3 + 0.00231481481481481*P*T**2*epsilon**2 - 0.00231481481481481*P*T**2*epsilon + 0.00154320987654321*R**3*epsilon**2 + 0.000385802469135802*R**3*epsilon - 0.000192901234567901*R**3 - 0.00462962962962963*R**2*S*epsilon**2 - 0.00115740740740741*R**2*S*epsilon + 0.000578703703703704*R**2*S - 0.00308641975308642*R**2*T*epsilon**3 - 0.00462962962962963*R**2*T*epsilon**2 - 0.000385802469135802*R**2*T*epsilon + 0.000578703703703704*R**2*T + 0.00462962962962963*R*S**2*epsilon**2 + 0.00115740740740741*R*S**2*epsilon - 0.000578703703703704*R*S**2 + 0.00617283950617284*R*S*T*epsilon**3 + 0.00925925925925926*R*S*T*epsilon**2 + 0.000771604938271605*R*S*T*epsilon - 0.00115740740740741*R*S*T + 0.00771604938271605*R*T**2*epsilon**3 + 0.00385802469135802*R*T**2*epsilon**2 - 0.000771604938271605*R*T**2*epsilon - 0.000385802469135802*R*T**2 - 0.00154320987654321*S**3*epsilon**2 - 0.000385802469135802*S**3*epsilon + 0.000192901234567901*S**3 - 0.00308641975308642*S**2*T*epsilon**3 - 0.00462962962962963*S**2*T*epsilon**2 - 0.000385802469135802*S**2*T*epsilon + 0.000578703703703704*S**2*T - 0.00771604938271605*S*T**2*epsilon**3 - 0.00385802469135802*S*T**2*epsilon**2 + 0.000771604938271605*S*T**2*epsilon + 0.000385802469135802*S*T**2 - 0.00462962962962963*T**3*epsilon**3 - 0.000771604938271605*T**3*epsilon**2 + 0.000771604938271605*T**3*epsilon) + N_G*UGA_star**4*UgA_star*(-0.00925925925925926*P**3*epsilon**3 - 0.00154320987654321*P**3*epsilon**2 + 0.00154320987654321*P**3*epsilon - 0.00308641975308642*P**2*R*epsilon**3 - 0.00617283950617284*P**2*R*epsilon**2 - 0.000771604938271605*P**2*R*epsilon + 0.000771604938271605*P**2*R + 0.00925925925925926*P**2*S*epsilon**3 + 0.00694444444444444*P**2*S*epsilon**2 - 0.000385802469135802*P**2*S*epsilon - 0.000771604938271605*P**2*S + 0.0231481481481481*P**2*T*epsilon**3 + 0.00385802469135802*P**2*T*epsilon**2 - 0.00385802469135802*P**2*T*epsilon - 0.00154320987654321*P*R**2*epsilon**2 - 0.000771604938271605*P*R**2*epsilon + 0.00308641975308642*P*R*S*epsilon**3 + 0.00694444444444444*P*R*S*epsilon**2 + 0.00154320987654321*P*R*S*epsilon - 0.000578703703703704*P*R*S + 0.00462962962962963*P*R*T*epsilon**3 + 0.00925925925925926*P*R*T*epsilon**2 + 0.00115740740740741*P*R*T*epsilon - 0.00115740740740741*P*R*T - 0.00308641975308642*P*S**2*epsilon**3 - 0.00540123456790123*P*S**2*epsilon**2 - 0.000771604938271605*P*S**2*epsilon + 0.000578703703703704*P*S**2 - 0.0169753086419753*P*S*T*epsilon**3 - 0.0108024691358025*P*S*T*epsilon**2 + 0.00115740740740741*P*S*T*epsilon + 0.00115740740740741*P*S*T - 0.0185185185185185*P*T**2*epsilon**3 - 0.00308641975308642*P*T**2*epsilon**2 + 0.00308641975308642*P*T**2*epsilon + 0.00154320987654321*R**2*S*epsilon**2 + 0.000385802469135802*R**2*S*epsilon - 0.000192901234567901*R**2*S + 0.000771604938271605*R**2*T*epsilon**2 + 0.000385802469135802*R**2*T*epsilon - 0.00308641975308642*R*S**2*epsilon**2 - 0.000771604938271605*R*S**2*epsilon + 0.000385802469135802*R*S**2 - 0.00308641975308642*R*S*T*epsilon**3 - 0.00540123456790123*R*S*T*epsilon**2 - 0.000771604938271605*R*S*T*epsilon + 0.000578703703703704*R*S*T - 0.00154320987654321*R*T**2*epsilon**3 - 0.00308641975308642*R*T**2*epsilon**2 - 0.000385802469135802*R*T**2*epsilon + 0.000385802469135802*R*T**2 + 0.00154320987654321*S**3*epsilon**2 + 0.000385802469135802*S**3*epsilon - 0.000192901234567901*S**3 + 0.00308641975308642*S**2*T*epsilon**3 + 0.00462962962962963*S**2*T*epsilon**2 + 0.000385802469135802*S**2*T*epsilon - 0.000578703703703704*S**2*T + 0.00771604938271605*S*T**2*epsilon**3 + 0.00385802469135802*S*T**2*epsilon**2 - 0.000771604938271605*S*T**2*epsilon - 0.000385802469135802*S*T**2 + 0.00462962962962963*T**3*epsilon**3 + 0.000771604938271605*T**3*epsilon**2 - 0.000771604938271605*T**3*epsilon) + N_G*UGA_star**4*(0.00154320987654321*M**2*P*epsilon**3 - 0.000771604938271605*M**2*P*epsilon**2 - 0.000385802469135802*M**2*P*epsilon + 0.000192901234567901*M**2*P + 0.00154320987654321*M**2*R*epsilon**3 - 0.000771604938271605*M**2*R*epsilon**2 - 0.000385802469135802*M**2*R*epsilon + 0.000192901234567901*M**2*R - 0.00154320987654321*M**2*S*epsilon**3 + 0.000771604938271605*M**2*S*epsilon**2 + 0.000385802469135802*M**2*S*epsilon - 0.000192901234567901*M**2*S - 0.00154320987654321*M**2*T*epsilon**3 + 0.000771604938271605*M**2*T*epsilon**2 + 0.000385802469135802*M**2*T*epsilon - 0.000192901234567901*M**2*T - 0.00308641975308642*M*P**2*UGA_prime_star*epsilon**3 + 0.000771604938271605*M*P**2*UGA_prime_star*epsilon + 0.0262345679012346*M*P**2*epsilon**3 + 0.0123456790123457*M*P**2*epsilon**2 - 0.00192901234567901*M*P**2*epsilon - 0.000771604938271605*M*P**2 - 0.00308641975308642*M*P*R*UGA_prime_star*epsilon**3 - 0.00154320987654321*M*P*R*UGA_prime_star*epsilon**2 + 0.000771604938271605*M*P*R*UGA_prime_star*epsilon + 0.000385802469135802*M*P*R*UGA_prime_star + 0.0401234567901235*M*P*R*epsilon**3 + 0.0169753086419753*M*P*R*epsilon**2 - 0.000771604938271605*M*P*R*epsilon + 0.000385802469135802*M*P*R + 0.00308641975308642*M*P*S*UGA_prime_star*epsilon**3 + 0.00154320987654321*M*P*S*UGA_prime_star*epsilon**2 - 0.000771604938271605*M*P*S*UGA_prime_star*epsilon - 0.000385802469135802*M*P*S*UGA_prime_star - 0.0493827160493827*M*P*S*epsilon**3 - 0.0169753086419753*M*P*S*epsilon**2 + 0.00308641975308642*M*P*S*epsilon - 0.000385802469135802*M*P*S + 0.00617283950617284*M*P*T*UGA_prime_star*epsilon**3 - 0.00154320987654321*M*P*T*UGA_prime_star*epsilon - 0.0462962962962963*M*P*T*epsilon**3 - 0.0231481481481481*M*P*T*epsilon**2 + 0.00231481481481481*M*P*T*epsilon + 0.00115740740740741*M*P*T - 0.00154320987654321*M*R**2*UGA_prime_star*epsilon**2 + 0.000385802469135802*M*R**2*UGA_prime_star + 0.00617283950617284*M*R**2*epsilon**3 + 0.00848765432098765*M*R**2*epsilon**2 + 0.00308641975308642*M*R**2*epsilon + 0.000192901234567901*M*R**2 + 0.00308641975308642*M*R*S*UGA_prime_star*epsilon**2 - 0.000771604938271605*M*R*S*UGA_prime_star - 0.0246913580246914*M*R*S*epsilon**3 - 0.0154320987654321*M*R*S*epsilon**2 - 0.00308641975308642*M*R*S*epsilon - 0.000771604938271605*M*R*S + 0.00308641975308642*M*R*T*UGA_prime_star*epsilon**3 + 0.00154320987654321*M*R*T*UGA_prime_star*epsilon**2 - 0.000771604938271605*M*R*T*UGA_prime_star*epsilon - 0.000385802469135802*M*R*T*UGA_prime_star - 0.0308641975308642*M*R*T*epsilon**3 - 0.0169753086419753*M*R*T*epsilon**2 - 0.00154320987654321*M*R*T*epsilon - 0.000385802469135802*M*R*T - 0.00154320987654321*M*S**2*UGA_prime_star*epsilon**2 + 0.000385802469135802*M*S**2*UGA_prime_star + 0.0185185185185185*M*S**2*epsilon**3 + 0.00694444444444444*M*S**2*epsilon**2 + 0.000578703703703704*M*S**2 - 0.00308641975308642*M*S*T*UGA_prime_star*epsilon**3 - 0.00154320987654321*M*S*T*UGA_prime_star*epsilon**2 + 0.000771604938271605*M*S*T*UGA_prime_star*epsilon + 0.000385802469135802*M*S*T*UGA_prime_star + 0.0401234567901235*M*S*T*epsilon**3 + 0.0169753086419753*M*S*T*epsilon**2 - 0.000771604938271605*M*S*T*epsilon + 0.000385802469135802*M*S*T - 0.00308641975308642*M*T**2*UGA_prime_star*epsilon**3 + 0.000771604938271605*M*T**2*UGA_prime_star*epsilon + 0.0200617283950617*M*T**2*epsilon**3 + 0.0108024691358025*M*T**2*epsilon**2 - 0.000385802469135802*M*T**2*epsilon - 0.000385802469135802*M*T**2 - 0.0648148148148148*P**3*epsilon**3 - 0.0455246913580247*P**3*epsilon**2 - 0.00655864197530864*P**3*epsilon + 0.00617283950617284*P**2*R*UGA_prime_star*epsilon**3 - 0.00154320987654321*P**2*R*UGA_prime_star*epsilon - 0.0632716049382716*P**2*R*epsilon**3 - 0.0478395061728395*P**2*R*epsilon**2 - 0.0111882716049383*P**2*R*epsilon - 0.00154320987654321*P**2*R - 0.00308641975308642*P**2*S*UGA_prime_star*epsilon**3 + 0.000771604938271605*P**2*S*UGA_prime_star*epsilon + 0.106481481481481*P**2*S*epsilon**3 + 0.0817901234567901*P**2*S*epsilon**2 + 0.0189043209876543*P**2*S*epsilon + 0.00231481481481481*P**2*S + 0.12962962962963*P**2*T*epsilon**3 + 0.0910493827160494*P**2*T*epsilon**2 + 0.0131172839506173*P**2*T*epsilon + 0.00308641975308642*P*R**2*UGA_prime_star*epsilon**2 - 0.000771604938271605*P*R**2*UGA_prime_star - 0.00617283950617284*P*R**2*epsilon**3 - 0.00771604938271605*P*R**2*epsilon**2 - 0.00270061728395062*P*R**2*epsilon - 0.000192901234567901*P*R**2 - 0.00462962962962963*P*R*S*UGA_prime_star*epsilon**2 + 0.00115740740740741*P*R*S*UGA_prime_star + 0.037037037037037*P*R*S*epsilon**3 + 0.0493827160493827*P*R*S*epsilon**2 + 0.0177469135802469*P*R*S*epsilon + 0.00115740740740741*P*R*S - 0.00925925925925926*P*R*T*UGA_prime_star*epsilon**3 + 0.00231481481481481*P*R*T*UGA_prime_star*epsilon + 0.0632716049382716*P*R*T*epsilon**3 + 0.0478395061728395*P*R*T*epsilon**2 + 0.0111882716049383*P*R*T*epsilon + 0.00154320987654321*P*R*T + 0.00154320987654321*P*S**2*UGA_prime_star*epsilon**2 - 0.000385802469135802*P*S**2*UGA_prime_star - 0.037037037037037*P*S**2*epsilon**3 - 0.0501543209876543*P*S**2*epsilon**2 - 0.0181327160493827*P*S**2*epsilon - 0.00115740740740741*P*S**2 + 0.00308641975308642*P*S*T*UGA_prime_star*epsilon**3 - 0.000771604938271605*P*S*T*UGA_prime_star*epsilon - 0.128086419753086*P*S*T*epsilon**3 - 0.0987654320987654*P*S*T*epsilon**2 - 0.0227623456790123*P*S*T*epsilon - 0.00270061728395062*P*S*T - 0.0787037037037037*P*T**2*epsilon**3 - 0.054783950617284*P*T**2*epsilon**2 - 0.00771604938271605*P*T**2*epsilon - 0.00154320987654321*R**2*T*UGA_prime_star*epsilon**2 + 0.000385802469135802*R**2*T*UGA_prime_star - 0.00925925925925926*R*S**2*epsilon**2 - 0.00462962962962963*R*S**2*epsilon + 0.00154320987654321*R*S*T*UGA_prime_star*epsilon**2 - 0.000385802469135802*R*S*T*UGA_prime_star - 0.0123456790123457*R*S*T*epsilon**3 - 0.0169753086419753*R*S*T*epsilon**2 - 0.00617283950617284*R*S*T*epsilon - 0.000385802469135802*R*S*T + 0.00308641975308642*R*T**2*UGA_prime_star*epsilon**3 - 0.000771604938271605*R*T**2*UGA_prime_star*epsilon - 0.0108024691358025*R*T**2*epsilon**3 - 0.00848765432098765*R*T**2*epsilon**2 - 0.00192901234567901*R*T**2*epsilon - 0.000192901234567901*R*T**2 + 0.00925925925925926*S**3*epsilon**2 + 0.00462962962962963*S**3*epsilon + 0.0185185185185185*S**2*T*epsilon**3 + 0.025462962962963*S**2*T*epsilon**2 + 0.00925925925925926*S**2*T*epsilon + 0.000578703703703704*S**2*T + 0.0324074074074074*S*T**2*epsilon**3 + 0.025462962962963*S*T**2*epsilon**2 + 0.00578703703703704*S*T**2*epsilon + 0.000578703703703704*S*T**2 + 0.0138888888888889*T**3*epsilon**3 + 0.00925925925925926*T**3*epsilon**2 + 0.00115740740740741*T**3*epsilon) + N_G*UGA_star**3*UgA_star**2*(0.00771604938271605*M*P**2*epsilon**3 - 0.00154320987654321*M*P**2*epsilon**2 - 0.00192901234567901*M*P**2*epsilon + 0.000385802469135802*M*P**2 + 0.0108024691358025*M*P*R*epsilon**3 - 0.000771604938271605*M*P*R*epsilon**2 - 0.00270061728395062*M*P*R*epsilon + 0.000192901234567901*M*P*R - 0.0108024691358025*M*P*S*epsilon**3 + 0.000771604938271605*M*P*S*epsilon**2 + 0.00270061728395062*M*P*S*epsilon - 0.000192901234567901*M*P*S - 0.0154320987654321*M*P*T*epsilon**3 + 0.00308641975308642*M*P*T*epsilon**2 + 0.00385802469135802*M*P*T*epsilon - 0.000771604938271605*M*P*T + 0.00308641975308642*M*R**2*epsilon**3 + 0.000771604938271605*M*R**2*epsilon**2 - 0.000771604938271605*M*R**2*epsilon - 0.000192901234567901*M*R**2 - 0.00617283950617284*M*R*S*epsilon**3 - 0.00154320987654321*M*R*S*epsilon**2 + 0.00154320987654321*M*R*S*epsilon + 0.000385802469135802*M*R*S - 0.0108024691358025*M*R*T*epsilon**3 + 0.000771604938271605*M*R*T*epsilon**2 + 0.00270061728395062*M*R*T*epsilon - 0.000192901234567901*M*R*T + 0.00308641975308642*M*S**2*epsilon**3 + 0.000771604938271605*M*S**2*epsilon**2 - 0.000771604938271605*M*S**2*epsilon - 0.000192901234567901*M*S**2 + 0.0108024691358025*M*S*T*epsilon**3 - 0.000771604938271605*M*S*T*epsilon**2 - 0.00270061728395062*M*S*T*epsilon + 0.000192901234567901*M*S*T + 0.00771604938271605*M*T**2*epsilon**3 - 0.00154320987654321*M*T**2*epsilon**2 - 0.00192901234567901*M*T**2*epsilon + 0.000385802469135802*M*T**2 - 0.0185185185185185*P**3*epsilon**3 - 0.00308641975308642*P**3*epsilon**2 + 0.00308641975308642*P**3*epsilon - 0.0277777777777778*P**2*R*epsilon**3 - 0.00925925925925926*P**2*R*epsilon**2 + 0.00385802469135802*P**2*R*epsilon + 0.000771604938271605*P**2*R + 0.029320987654321*P**2*S*epsilon**3 + 0.0123456790123457*P**2*S*epsilon**2 - 0.00347222222222222*P**2*S*epsilon - 0.00115740740740741*P**2*S + 0.0462962962962963*P**2*T*epsilon**3 + 0.00771604938271605*P**2*T*epsilon**2 - 0.00771604938271605*P**2*T*epsilon - 0.00617283950617284*P*R**2*epsilon**3 - 0.00771604938271605*P*R**2*epsilon**2 + 0.00115740740740741*P*R**2 + 0.0154320987654321*P*R*S*epsilon**3 + 0.0208333333333333*P*R*S*epsilon**2 + 0.000771604938271605*P*R*S*epsilon - 0.00289351851851852*P*R*S + 0.0416666666666667*P*R*T*epsilon**3 + 0.0138888888888889*P*R*T*epsilon**2 - 0.00578703703703704*P*R*T*epsilon - 0.00115740740740741*P*R*T - 0.00925925925925926*P*S**2*epsilon**3 - 0.0131172839506173*P*S**2*epsilon**2 - 0.000771604938271605*P*S**2*epsilon + 0.00173611111111111*P*S**2 - 0.0447530864197531*P*S*T*epsilon**3 - 0.0200617283950617*P*S*T*epsilon**2 + 0.00501543209876543*P*S*T*epsilon + 0.00192901234567901*P*S*T - 0.037037037037037*P*T**2*epsilon**3 - 0.00617283950617284*P*T**2*epsilon**2 + 0.00617283950617284*P*T**2*epsilon + 0.00308641975308642*R**2*S*epsilon**2 + 0.000771604938271605*R**2*S*epsilon - 0.000385802469135802*R**2*S + 0.00308641975308642*R**2*T*epsilon**3 + 0.00385802469135802*R**2*T*epsilon**2 - 0.000578703703703704*R**2*T - 0.00617283950617284*R*S**2*epsilon**2 - 0.00154320987654321*R*S**2*epsilon + 0.000771604938271605*R*S**2 - 0.00925925925925926*R*S*T*epsilon**3 - 0.0131172839506173*R*S*T*epsilon**2 - 0.000771604938271605*R*S*T*epsilon + 0.00173611111111111*R*S*T - 0.0138888888888889*R*T**2*epsilon**3 - 0.00462962962962963*R*T**2*epsilon**2 + 0.00192901234567901*R*T**2*epsilon + 0.000385802469135802*R*T**2 + 0.00308641975308642*S**3*epsilon**2 + 0.000771604938271605*S**3*epsilon - 0.000385802469135802*S**3 + 0.00617283950617284*S**2*T*epsilon**3 + 0.00925925925925926*S**2*T*epsilon**2 + 0.000771604938271605*S**2*T*epsilon - 0.00115740740740741*S**2*T + 0.0154320987654321*S*T**2*epsilon**3 + 0.00771604938271605*S*T**2*epsilon**2 - 0.00154320987654321*S*T**2*epsilon - 0.000771604938271605*S*T**2 + 0.00925925925925926*T**3*epsilon**3 + 0.00154320987654321*T**3*epsilon**2 - 0.00154320987654321*T**3*epsilon) + N_G*UGA_star**3*UgA_star*(-0.0154320987654321*M*P**2*epsilon**3 + 0.00308641975308642*M*P**2*epsilon**2 + 0.00385802469135802*M*P**2*epsilon - 0.000771604938271605*M*P**2 - 0.00308641975308642*M*P*R*epsilon**3 - 0.00308641975308642*M*P*R*epsilon**2 + 0.000771604938271605*M*P*R*epsilon + 0.000771604938271605*M*P*R + 0.0123456790123457*M*P*S*epsilon**3 + 0.000771604938271605*M*P*S*epsilon**2 - 0.00308641975308642*M*P*S*epsilon - 0.000192901234567901*M*P*S + 0.0231481481481481*M*P*T*epsilon**3 - 0.00462962962962963*M*P*T*epsilon**2 - 0.00578703703703704*M*P*T*epsilon + 0.00115740740740741*M*P*T + 0.00308641975308642*M*R*S*epsilon**3 + 0.000771604938271605*M*R*S*epsilon**2 - 0.000771604938271605*M*R*S*epsilon - 0.000192901234567901*M*R*S + 0.00154320987654321*M*R*T*epsilon**3 + 0.00154320987654321*M*R*T*epsilon**2 - 0.000385802469135802*M*R*T*epsilon - 0.000385802469135802*M*R*T - 0.00308641975308642*M*S**2*epsilon**3 - 0.000771604938271605*M*S**2*epsilon**2 + 0.000771604938271605*M*S**2*epsilon + 0.000192901234567901*M*S**2 - 0.0108024691358025*M*S*T*epsilon**3 + 0.000771604938271605*M*S*T*epsilon**2 + 0.00270061728395062*M*S*T*epsilon - 0.000192901234567901*M*S*T - 0.00771604938271605*M*T**2*epsilon**3 + 0.00154320987654321*M*T**2*epsilon**2 + 0.00192901234567901*M*T**2*epsilon - 0.000385802469135802*M*T**2 + 0.037037037037037*P**3*epsilon**3 + 0.00617283950617284*P**3*epsilon**2 - 0.00617283950617284*P**3*epsilon + 0.00617283950617284*P**2*R*epsilon**3 + 0.0123456790123457*P**2*R*epsilon**2 + 0.00154320987654321*P**2*R*epsilon - 0.00154320987654321*P**2*R - 0.0339506172839506*P**2*S*epsilon**3 - 0.0216049382716049*P**2*S*epsilon**2 + 0.00231481481481481*P**2*S*epsilon + 0.00231481481481481*P**2*S - 0.0740740740740741*P**2*T*epsilon**3 - 0.0123456790123457*P**2*T*epsilon**2 + 0.0123456790123457*P**2*T*epsilon - 0.00617283950617284*P*R*S*epsilon**3 - 0.0108024691358025*P*R*S*epsilon**2 - 0.00154320987654321*P*R*S*epsilon + 0.00115740740740741*P*R*S - 0.00617283950617284*P*R*T*epsilon**3 - 0.0123456790123457*P*R*T*epsilon**2 - 0.00154320987654321*P*R*T*epsilon + 0.00154320987654321*P*R*T + 0.00925925925925926*P*S**2*epsilon**3 + 0.0146604938271605*P*S**2*epsilon**2 + 0.00154320987654321*P*S**2*epsilon - 0.00173611111111111*P*S**2 + 0.0478395061728395*P*S*T*epsilon**3 + 0.0262345679012346*P*S*T*epsilon**2 - 0.00424382716049383*P*S*T*epsilon - 0.00270061728395062*P*S*T + 0.0462962962962963*P*T**2*epsilon**3 + 0.00771604938271605*P*T**2*epsilon**2 - 0.00771604938271605*P*T**2*epsilon + 0.00308641975308642*R*S**2*epsilon**2 + 0.000771604938271605*R*S**2*epsilon - 0.000385802469135802*R*S**2 + 0.00308641975308642*R*S*T*epsilon**3 + 0.00540123456790123*R*S*T*epsilon**2 + 0.000771604938271605*R*S*T*epsilon - 0.000578703703703704*R*S*T + 0.00154320987654321*R*T**2*epsilon**3 + 0.00308641975308642*R*T**2*epsilon**2 + 0.000385802469135802*R*T**2*epsilon - 0.000385802469135802*R*T**2 - 0.00308641975308642*S**3*epsilon**2 - 0.000771604938271605*S**3*epsilon + 0.000385802469135802*S**3 - 0.00617283950617284*S**2*T*epsilon**3 - 0.00925925925925926*S**2*T*epsilon**2 - 0.000771604938271605*S**2*T*epsilon + 0.00115740740740741*S**2*T - 0.0154320987654321*S*T**2*epsilon**3 - 0.00771604938271605*S*T**2*epsilon**2 + 0.00154320987654321*S*T**2*epsilon + 0.000771604938271605*S*T**2 - 0.00925925925925926*T**3*epsilon**3 - 0.00154320987654321*T**3*epsilon**2 + 0.00154320987654321*T**3*epsilon) + N_G*UGA_star**3*(-0.00308641975308642*M**2*P*UGA_prime_star*epsilon**3 + 0.00154320987654321*M**2*P*UGA_prime_star*epsilon**2 + 0.000771604938271605*M**2*P*UGA_prime_star*epsilon - 0.000385802469135802*M**2*P*UGA_prime_star + 0.00308641975308642*M**2*P*epsilon**3 + 0.00308641975308642*M**2*P*epsilon**2 - 0.000771604938271605*M**2*P*epsilon - 0.000771604938271605*M**2*P - 0.00308641975308642*M**2*R*UGA_prime_star*epsilon**3 + 0.00154320987654321*M**2*R*UGA_prime_star*epsilon**2 + 0.000771604938271605*M**2*R*UGA_prime_star*epsilon - 0.000385802469135802*M**2*R*UGA_prime_star + 0.00925925925925926*M**2*R*epsilon**3 - 0.00231481481481481*M**2*R*epsilon + 0.00308641975308642*M**2*S*UGA_prime_star*epsilon**3 - 0.00154320987654321*M**2*S*UGA_prime_star*epsilon**2 - 0.000771604938271605*M**2*S*UGA_prime_star*epsilon + 0.000385802469135802*M**2*S*UGA_prime_star - 0.00771604938271605*M**2*S*epsilon**3 - 0.000771604938271605*M**2*S*epsilon**2 + 0.00192901234567901*M**2*S*epsilon + 0.000192901234567901*M**2*S + 0.00308641975308642*M**2*T*UGA_prime_star*epsilon**3 - 0.00154320987654321*M**2*T*UGA_prime_star*epsilon**2 - 0.000771604938271605*M**2*T*UGA_prime_star*epsilon + 0.000385802469135802*M**2*T*UGA_prime_star - 0.00462962962962963*M**2*T*epsilon**3 - 0.00231481481481481*M**2*T*epsilon**2 + 0.00115740740740741*M**2*T*epsilon + 0.000578703703703704*M**2*T + 0.00617283950617284*M*P**2*UGA_prime_star*epsilon**3 - 0.00154320987654321*M*P**2*UGA_prime_star*epsilon - 0.066358024691358*M*P**2*epsilon**3 - 0.0432098765432099*M*P**2*epsilon**2 - 0.00192901234567901*M*P**2*epsilon + 0.00154320987654321*M*P**2 + 0.0185185185185185*M*P*R*UGA_prime_star*epsilon**3 - 0.00308641975308642*M*P*R*UGA_prime_star*epsilon**2 - 0.00462962962962963*M*P*R*UGA_prime_star*epsilon + 0.000771604938271605*M*P*R*UGA_prime_star - 0.0509259259259259*M*P*R*epsilon**3 - 0.0324074074074074*M*P*R*epsilon**2 - 0.00578703703703704*M*P*R*epsilon - 0.00115740740740741*M*P*R - 0.0154320987654321*M*P*S*UGA_prime_star*epsilon**3 + 0.00385802469135802*M*P*S*UGA_prime_star*epsilon + 0.0910493827160494*M*P*S*epsilon**3 + 0.0493827160493827*M*P*S*epsilon**2 + 0.00501543209876543*M*P*S*epsilon + 0.00154320987654321*M*P*S - 0.00925925925925926*M*P*T*UGA_prime_star*epsilon**3 + 0.00231481481481481*M*P*T*UGA_prime_star*epsilon + 0.0941358024691358*M*P*T*epsilon**3 + 0.0617283950617284*M*P*T*epsilon**2 + 0.00424382716049383*M*P*T*epsilon - 0.00154320987654321*M*P*T + 0.00308641975308642*M*R**2*UGA_prime_star*epsilon**2 - 0.000771604938271605*M*R**2*UGA_prime_star - 0.00771604938271605*M*R*S*UGA_prime_star*epsilon**2 + 0.00192901234567901*M*R*S*UGA_prime_star + 0.0123456790123457*M*R*S*epsilon**3 + 0.0169753086419753*M*R*S*epsilon**2 + 0.00617283950617284*M*R*S*epsilon + 0.000385802469135802*M*R*S - 0.0123456790123457*M*R*T*UGA_prime_star*epsilon**3 + 0.00154320987654321*M*R*T*UGA_prime_star*epsilon**2 + 0.00308641975308642*M*R*T*UGA_prime_star*epsilon - 0.000385802469135802*M*R*T*UGA_prime_star + 0.0216049382716049*M*R*T*epsilon**3 + 0.0169753086419753*M*R*T*epsilon**2 + 0.00385802469135802*M*R*T*epsilon + 0.000385802469135802*M*R*T + 0.00462962962962963*M*S**2*UGA_prime_star*epsilon**2 - 0.00115740740740741*M*S**2*UGA_prime_star - 0.0185185185185185*M*S**2*epsilon**3 - 0.0162037037037037*M*S**2*epsilon**2 - 0.00462962962962963*M*S**2*epsilon - 0.000578703703703704*M*S**2 + 0.00925925925925926*M*S*T*UGA_prime_star*epsilon**3 + 0.00154320987654321*M*S*T*UGA_prime_star*epsilon**2 - 0.00231481481481481*M*S*T*UGA_prime_star*epsilon - 0.000385802469135802*M*S*T*UGA_prime_star - 0.0524691358024691*M*S*T*epsilon**3 - 0.0339506172839506*M*S*T*epsilon**2 - 0.00540123456790123*M*S*T*epsilon - 0.000771604938271605*M*S*T + 0.00308641975308642*M*T**2*UGA_prime_star*epsilon**3 - 0.000771604938271605*M*T**2*UGA_prime_star*epsilon - 0.0308641975308642*M*T**2*epsilon**3 - 0.0192901234567901*M*T**2*epsilon**2 - 0.00154320987654321*M*T**2*epsilon + 0.000192901234567901*M*T**2 + 0.0740740740740741*P**3*epsilon**3 + 0.058641975308642*P**3*epsilon**2 + 0.0108024691358025*P**3*epsilon - 0.0185185185185185*P**2*R*UGA_prime_star*epsilon**3 + 0.00462962962962963*P**2*R*UGA_prime_star*epsilon + 0.0401234567901235*P**2*R*epsilon**3 + 0.0277777777777778*P**2*R*epsilon**2 + 0.00694444444444444*P**2*R*epsilon + 0.00154320987654321*P**2*R + 0.0123456790123457*P**2*S*UGA_prime_star*epsilon**3 - 0.00308641975308642*P**2*S*UGA_prime_star*epsilon - 0.103395061728395*P**2*S*epsilon**3 - 0.0756172839506173*P**2*S*epsilon**2 - 0.0181327160493827*P**2*S*epsilon - 0.00308641975308642*P**2*S - 0.111111111111111*P**2*T*epsilon**3 - 0.087962962962963*P**2*T*epsilon**2 - 0.0162037037037037*P**2*T*epsilon - 0.00308641975308642*P*R**2*UGA_prime_star*epsilon**2 + 0.000771604938271605*P*R**2*UGA_prime_star + 0.00925925925925926*P*R*S*UGA_prime_star*epsilon**2 - 0.00231481481481481*P*R*S*UGA_prime_star - 0.0123456790123457*P*R*S*epsilon**3 - 0.0154320987654321*P*R*S*epsilon**2 - 0.00540123456790123*P*R*S*epsilon - 0.000385802469135802*P*R*S + 0.0185185185185185*P*R*T*UGA_prime_star*epsilon**3 - 0.00462962962962963*P*R*T*UGA_prime_star*epsilon - 0.0200617283950617*P*R*T*epsilon**3 - 0.0138888888888889*P*R*T*epsilon**2 - 0.00347222222222222*P*R*T*epsilon - 0.000771604938271605*P*R*T - 0.00462962962962963*P*S**2*UGA_prime_star*epsilon**2 + 0.00115740740740741*P*S**2*UGA_prime_star + 0.0246913580246914*P*S**2*epsilon**3 + 0.0324074074074074*P*S**2*epsilon**2 + 0.0115740740740741*P*S**2*epsilon + 0.000771604938271605*P*S**2 - 0.00925925925925926*P*S*T*UGA_prime_star*epsilon**3 + 0.00231481481481481*P*S*T*UGA_prime_star*epsilon + 0.0833333333333333*P*S*T*epsilon**3 + 0.0617283950617284*P*S*T*epsilon**2 + 0.0146604938271605*P*S*T*epsilon + 0.00231481481481481*P*S*T + 0.0462962962962963*P*T**2*epsilon**3 + 0.0354938271604938*P*T**2*epsilon**2 + 0.00617283950617284*P*T**2*epsilon - 0.00154320987654321*R*S*T*UGA_prime_star*epsilon**2 + 0.000385802469135802*R*S*T*UGA_prime_star - 0.00308641975308642*R*T**2*UGA_prime_star*epsilon**3 + 0.000771604938271605*R*T**2*UGA_prime_star*epsilon - 0.00308641975308642*S**3*epsilon**2 - 0.00154320987654321*S**3*epsilon - 0.00617283950617284*S**2*T*epsilon**3 - 0.00848765432098765*S**2*T*epsilon**2 - 0.00308641975308642*S**2*T*epsilon - 0.000192901234567901*S**2*T - 0.0108024691358025*S*T**2*epsilon**3 - 0.00848765432098765*S*T**2*epsilon**2 - 0.00192901234567901*S*T**2*epsilon - 0.000192901234567901*S*T**2 - 0.00462962962962963*T**3*epsilon**3 - 0.00308641975308642*T**3*epsilon**2 - 0.000385802469135802*T**3*epsilon) + N_G*UGA_star**2*UgA_star**2*(0.00308641975308642*M**2*P*epsilon**3 - 0.00154320987654321*M**2*P*epsilon**2 - 0.000771604938271605*M**2*P*epsilon + 0.000385802469135802*M**2*P + 0.00308641975308642*M**2*R*epsilon**3 - 0.00154320987654321*M**2*R*epsilon**2 - 0.000771604938271605*M**2*R*epsilon + 0.000385802469135802*M**2*R - 0.00308641975308642*M**2*S*epsilon**3 + 0.00154320987654321*M**2*S*epsilon**2 + 0.000771604938271605*M**2*S*epsilon - 0.000385802469135802*M**2*S - 0.00308641975308642*M**2*T*epsilon**3 + 0.00154320987654321*M**2*T*epsilon**2 + 0.000771604938271605*M**2*T*epsilon - 0.000385802469135802*M**2*T - 0.0246913580246914*M*P**2*epsilon**3 + 0.00154320987654321*M*P**2*epsilon**2 + 0.00540123456790123*M*P**2*epsilon - 0.000771604938271605*M*P**2 - 0.0324074074074074*M*P*R*epsilon**3 + 0.00655864197530864*M*P*R*epsilon - 0.000771604938271605*M*P*R + 0.0339506172839506*M*P*S*epsilon**3 + 0.00154320987654321*M*P*S*epsilon**2 - 0.00694444444444444*M*P*S*epsilon + 0.000385802469135802*M*P*S + 0.0416666666666667*M*P*T*epsilon**3 - 0.00154320987654321*M*P*T*epsilon**2 - 0.00887345679012346*M*P*T*epsilon + 0.00115740740740741*M*P*T - 0.00308641975308642*M*R**2*epsilon**3 - 0.00385802469135802*M*R**2*epsilon**2 + 0.000578703703703704*M*R**2 + 0.00925925925925926*M*R*S*epsilon**3 + 0.00848765432098765*M*R*S*epsilon**2 - 0.000771604938271605*M*R*S*epsilon - 0.00135030864197531*M*R*S + 0.0231481481481481*M*R*T*epsilon**3 + 0.00231481481481481*M*R*T*epsilon**2 - 0.00424382716049383*M*R*T*epsilon + 0.000192901234567901*M*R*T - 0.00617283950617284*M*S**2*epsilon**3 - 0.00462962962962963*M*S**2*epsilon**2 + 0.000771604938271605*M*S**2*epsilon + 0.000771604938271605*M*S**2 - 0.0246913580246914*M*S*T*epsilon**3 - 0.00385802469135802*M*S*T*epsilon**2 + 0.00462962962962963*M*S*T*epsilon + 0.000192901234567901*M*S*T - 0.0169753086419753*M*T**2*epsilon**3 + 0.00347222222222222*M*T**2*epsilon - 0.000385802469135802*M*T**2 + 0.0277777777777778*P**3*epsilon**3 + 0.00462962962962963*P**3*epsilon**2 - 0.00462962962962963*P**3*epsilon + 0.0385802469135802*P**2*R*epsilon**3 + 0.00771604938271605*P**2*R*epsilon**2 - 0.00655864197530864*P**2*R*epsilon - 0.000385802469135802*P**2*R - 0.0416666666666667*P**2*S*epsilon**3 - 0.0138888888888889*P**2*S*epsilon**2 + 0.00578703703703704*P**2*S*epsilon + 0.00115740740740741*P**2*S - 0.0555555555555556*P**2*T*epsilon**3 - 0.00925925925925926*P**2*T*epsilon**2 + 0.00925925925925926*P**2*T*epsilon + 0.00308641975308642*P*R**2*epsilon**3 + 0.00385802469135802*P*R**2*epsilon**2 - 0.000578703703703704*P*R**2 - 0.0123456790123457*P*R*S*epsilon**3 - 0.0154320987654321*P*R*S*epsilon**2 + 0.00231481481481481*P*R*S - 0.0385802469135802*P*R*T*epsilon**3 - 0.00771604938271605*P*R*T*epsilon**2 + 0.00655864197530864*P*R*T*epsilon + 0.000385802469135802*P*R*T + 0.00925925925925926*P*S**2*epsilon**3 + 0.0123456790123457*P*S**2*epsilon**2 + 0.000385802469135802*P*S**2*epsilon - 0.00173611111111111*P*S**2 + 0.0432098765432099*P*S*T*epsilon**3 + 0.0169753086419753*P*S*T*epsilon**2 - 0.00540123456790123*P*S*T*epsilon - 0.00154320987654321*P*S*T + 0.0324074074074074*P*T**2*epsilon**3 + 0.00540123456790123*P*T**2*epsilon**2 - 0.00540123456790123*P*T**2*epsilon + 0.00154320987654321*R*S**2*epsilon**2 + 0.000385802469135802*R*S**2*epsilon - 0.000192901234567901*R*S**2 + 0.00308641975308642*R*S*T*epsilon**3 + 0.00385802469135802*R*S*T*epsilon**2 - 0.000578703703703704*R*S*T + 0.00617283950617284*R*T**2*epsilon**3 + 0.000771604938271605*R*T**2*epsilon**2 - 0.00115740740740741*R*T**2*epsilon - 0.00154320987654321*S**3*epsilon**2 - 0.000385802469135802*S**3*epsilon + 0.000192901234567901*S**3 - 0.00308641975308642*S**2*T*epsilon**3 - 0.00462962962962963*S**2*T*epsilon**2 - 0.000385802469135802*S**2*T*epsilon + 0.000578703703703704*S**2*T - 0.00771604938271605*S*T**2*epsilon**3 - 0.00385802469135802*S*T**2*epsilon**2 + 0.000771604938271605*S*T**2*epsilon + 0.000385802469135802*S*T**2 - 0.00462962962962963*T**3*epsilon**3 - 0.000771604938271605*T**3*epsilon**2 + 0.000771604938271605*T**3*epsilon) + N_G*UGA_star**2*UgA_star*(-0.00617283950617284*M**2*P*epsilon**3 + 0.00308641975308642*M**2*P*epsilon**2 + 0.00154320987654321*M**2*P*epsilon - 0.000771604938271605*M**2*P + 0.00308641975308642*M**2*S*epsilon**3 - 0.00154320987654321*M**2*S*epsilon**2 - 0.000771604938271605*M**2*S*epsilon + 0.000385802469135802*M**2*S + 0.00308641975308642*M**2*T*epsilon**3 - 0.00154320987654321*M**2*T*epsilon**2 - 0.000771604938271605*M**2*T*epsilon + 0.000385802469135802*M**2*T + 0.0493827160493827*M*P**2*epsilon**3 - 0.00308641975308642*M*P**2*epsilon**2 - 0.0108024691358025*M*P**2*epsilon + 0.00154320987654321*M*P**2 + 0.00308641975308642*M*P*R*epsilon**3 + 0.00617283950617284*M*P*R*epsilon**2 + 0.000771604938271605*M*P*R*epsilon - 0.000771604938271605*M*P*R - 0.037037037037037*M*P*S*epsilon**3 - 0.00617283950617284*M*P*S*epsilon**2 + 0.00694444444444444*M*P*S*epsilon + 0.000385802469135802*M*P*S - 0.058641975308642*M*P*T*epsilon**3 + 0.00154320987654321*M*P*T*epsilon**2 + 0.0123456790123457*M*P*T*epsilon - 0.00154320987654321*M*P*T - 0.00308641975308642*M*R*S*epsilon**3 - 0.00385802469135802*M*R*S*epsilon**2 + 0.000578703703703704*M*R*S - 0.00154320987654321*M*R*T*epsilon**3 - 0.00308641975308642*M*R*T*epsilon**2 - 0.000385802469135802*M*R*T*epsilon + 0.000385802469135802*M*R*T + 0.00617283950617284*M*S**2*epsilon**3 + 0.00462962962962963*M*S**2*epsilon**2 - 0.000771604938271605*M*S**2*epsilon - 0.000771604938271605*M*S**2 + 0.0246913580246914*M*S*T*epsilon**3 + 0.00385802469135802*M*S*T*epsilon**2 - 0.00462962962962963*M*S*T*epsilon - 0.000192901234567901*M*S*T + 0.0169753086419753*M*T**2*epsilon**3 - 0.00347222222222222*M*T**2*epsilon + 0.000385802469135802*M*T**2 - 0.0555555555555556*P**3*epsilon**3 - 0.00925925925925926*P**3*epsilon**2 + 0.00925925925925926*P**3*epsilon - 0.00308641975308642*P**2*R*epsilon**3 - 0.00617283950617284*P**2*R*epsilon**2 - 0.000771604938271605*P**2*R*epsilon + 0.000771604938271605*P**2*R + 0.0462962962962963*P**2*S*epsilon**3 + 0.0231481481481481*P**2*S*epsilon**2 - 0.00462962962962963*P**2*S*epsilon - 0.00231481481481481*P**2*S + 0.0833333333333333*P**2*T*epsilon**3 + 0.0138888888888889*P**2*T*epsilon**2 - 0.0138888888888889*P**2*T*epsilon + 0.00308641975308642*P*R*S*epsilon**3 + 0.00385802469135802*P*R*S*epsilon**2 - 0.000578703703703704*P*R*S + 0.00154320987654321*P*R*T*epsilon**3 + 0.00308641975308642*P*R*T*epsilon**2 + 0.000385802469135802*P*R*T*epsilon - 0.000385802469135802*P*R*T - 0.00925925925925926*P*S**2*epsilon**3 - 0.0131172839506173*P*S**2*epsilon**2 - 0.000771604938271605*P*S**2*epsilon + 0.00173611111111111*P*S**2 - 0.0447530864197531*P*S*T*epsilon**3 - 0.0200617283950617*P*S*T*epsilon**2 + 0.00501543209876543*P*S*T*epsilon + 0.00192901234567901*P*S*T - 0.037037037037037*P*T**2*epsilon**3 - 0.00617283950617284*P*T**2*epsilon**2 + 0.00617283950617284*P*T**2*epsilon + 0.00154320987654321*S**3*epsilon**2 + 0.000385802469135802*S**3*epsilon - 0.000192901234567901*S**3 + 0.00308641975308642*S**2*T*epsilon**3 + 0.00462962962962963*S**2*T*epsilon**2 + 0.000385802469135802*S**2*T*epsilon - 0.000578703703703704*S**2*T + 0.00771604938271605*S*T**2*epsilon**3 + 0.00385802469135802*S*T**2*epsilon**2 - 0.000771604938271605*S*T**2*epsilon - 0.000385802469135802*S*T**2 + 0.00462962962962963*T**3*epsilon**3 + 0.000771604938271605*T**3*epsilon**2 - 0.000771604938271605*T**3*epsilon) + N_G*UGA_star**2*(-0.00154320987654321*M**3*epsilon**3 + 0.000771604938271605*M**3*epsilon**2 + 0.000385802469135802*M**3*epsilon - 0.000192901234567901*M**3 - 0.0216049382716049*M**2*P*epsilon**3 - 0.0146604938271605*M**2*P*epsilon**2 + 0.000771604938271605*M**2*P*epsilon + 0.00135030864197531*M**2*P + 0.00925925925925926*M**2*R*UGA_prime_star*epsilon**3 - 0.00462962962962963*M**2*R*UGA_prime_star*epsilon**2 - 0.00231481481481481*M**2*R*UGA_prime_star*epsilon + 0.00115740740740741*M**2*R*UGA_prime_star - 0.0108024691358025*M**2*R*epsilon**3 - 0.00848765432098765*M**2*R*epsilon**2 - 0.00192901234567901*M**2*R*epsilon - 0.000192901234567901*M**2*R - 0.00925925925925926*M**2*S*UGA_prime_star*epsilon**3 + 0.00462962962962963*M**2*S*UGA_prime_star*epsilon**2 + 0.00231481481481481*M**2*S*UGA_prime_star*epsilon - 0.00115740740740741*M**2*S*UGA_prime_star + 0.0200617283950617*M**2*S*epsilon**3 + 0.00848765432098765*M**2*S*epsilon**2 - 0.000385802469135802*M**2*S*epsilon + 0.000192901234567901*M**2*S + 0.0200617283950617*M**2*T*epsilon**3 + 0.0108024691358025*M**2*T*epsilon**2 - 0.000385802469135802*M**2*T*epsilon - 0.000385802469135802*M**2*T + 0.0709876543209877*M*P**2*epsilon**3 + 0.0601851851851852*M*P**2*epsilon**2 + 0.00925925925925926*M*P**2*epsilon - 0.00154320987654321*M*P**2 - 0.0277777777777778*M*P*R*UGA_prime_star*epsilon**3 + 0.00462962962962963*M*P*R*UGA_prime_star*epsilon**2 + 0.00694444444444444*M*P*R*UGA_prime_star*epsilon - 0.00115740740740741*M*P*R*UGA_prime_star + 0.0200617283950617*M*P*R*epsilon**3 + 0.0138888888888889*M*P*R*epsilon**2 + 0.00347222222222222*M*P*R*epsilon + 0.000771604938271605*M*P*R + 0.0277777777777778*M*P*S*UGA_prime_star*epsilon**3 - 0.00462962962962963*M*P*S*UGA_prime_star*epsilon**2 - 0.00694444444444444*M*P*S*UGA_prime_star*epsilon + 0.00115740740740741*M*P*S*UGA_prime_star - 0.0709876543209877*M*P*S*epsilon**3 - 0.0462962962962963*M*P*S*epsilon**2 - 0.00925925925925926*M*P*S*epsilon - 0.00192901234567901*M*P*S - 0.0725308641975309*M*P*T*epsilon**3 - 0.0570987654320988*M*P*T*epsilon**2 - 0.00887345679012346*M*P*T*epsilon + 0.000771604938271605*M*P*T + 0.00462962962962963*M*R*S*UGA_prime_star*epsilon**2 - 0.00115740740740741*M*R*S*UGA_prime_star + 0.00925925925925926*M*R*T*UGA_prime_star*epsilon**3 - 0.00231481481481481*M*R*T*UGA_prime_star*epsilon - 0.00462962962962963*M*S**2*UGA_prime_star*epsilon**2 + 0.00115740740740741*M*S**2*UGA_prime_star + 0.00617283950617284*M*S**2*epsilon**3 + 0.00848765432098765*M*S**2*epsilon**2 + 0.00308641975308642*M*S**2*epsilon + 0.000192901234567901*M*S**2 - 0.00925925925925926*M*S*T*UGA_prime_star*epsilon**3 + 0.00231481481481481*M*S*T*UGA_prime_star*epsilon + 0.0216049382716049*M*S*T*epsilon**3 + 0.0169753086419753*M*S*T*epsilon**2 + 0.00385802469135802*M*S*T*epsilon + 0.000385802469135802*M*S*T + 0.0138888888888889*M*T**2*epsilon**3 + 0.00925925925925926*M*T**2*epsilon**2 + 0.00115740740740741*M*T**2*epsilon - 0.0416666666666667*P**3*epsilon**3 - 0.0416666666666667*P**3*epsilon**2 - 0.0104166666666667*P**3*epsilon + 0.0185185185185185*P**2*R*UGA_prime_star*epsilon**3 - 0.00462962962962963*P**2*R*UGA_prime_star*epsilon - 0.00925925925925926*P**2*R*epsilon**3 - 0.00540123456790123*P**2*R*epsilon**2 - 0.00154320987654321*P**2*R*epsilon - 0.000578703703703704*P**2*R - 0.0185185185185185*P**2*S*UGA_prime_star*epsilon**3 + 0.00462962962962963*P**2*S*UGA_prime_star*epsilon + 0.0493827160493827*P**2*S*epsilon**3 + 0.033179012345679*P**2*S*epsilon**2 + 0.00848765432098765*P**2*S*epsilon + 0.00212191358024691*P**2*S + 0.0416666666666667*P**2*T*epsilon**3 + 0.0416666666666667*P**2*T*epsilon**2 + 0.0104166666666667*P**2*T*epsilon - 0.00462962962962963*P*R*S*UGA_prime_star*epsilon**2 + 0.00115740740740741*P*R*S*UGA_prime_star - 0.00925925925925926*P*R*T*UGA_prime_star*epsilon**3 + 0.00231481481481481*P*R*T*UGA_prime_star*epsilon + 0.00462962962962963*P*S**2*UGA_prime_star*epsilon**2 - 0.00115740740740741*P*S**2*UGA_prime_star - 0.00617283950617284*P*S**2*epsilon**3 - 0.00771604938271605*P*S**2*epsilon**2 - 0.00270061728395062*P*S**2*epsilon - 0.000192901234567901*P*S**2 + 0.00925925925925926*P*S*T*UGA_prime_star*epsilon**3 - 0.00231481481481481*P*S*T*UGA_prime_star*epsilon - 0.0200617283950617*P*S*T*epsilon**3 - 0.0138888888888889*P*S*T*epsilon**2 - 0.00347222222222222*P*S*T*epsilon - 0.000771604938271605*P*S*T - 0.00925925925925926*P*T**2*epsilon**3 - 0.00848765432098765*P*T**2*epsilon**2 - 0.00192901234567901*P*T**2*epsilon) + N_G*UGA_star*UgA_star**2*(-0.00771604938271605*M**2*P*epsilon**3 + 0.00154320987654321*M**2*P*epsilon**2 + 0.00192901234567901*M**2*P*epsilon - 0.000385802469135802*M**2*P - 0.00925925925925926*M**2*R*epsilon**3 + 0.00231481481481481*M**2*R*epsilon**2 + 0.00231481481481481*M**2*R*epsilon - 0.000578703703703704*M**2*R + 0.00925925925925926*M**2*S*epsilon**3 - 0.00231481481481481*M**2*S*epsilon**2 - 0.00231481481481481*M**2*S*epsilon + 0.000578703703703704*M**2*S + 0.00771604938271605*M**2*T*epsilon**3 - 0.00154320987654321*M**2*T*epsilon**2 - 0.00192901234567901*M**2*T*epsilon + 0.000385802469135802*M**2*T + 0.0262345679012346*M*P**2*epsilon**3 + 0.00154320987654321*M*P**2*epsilon**2 - 0.00501543209876543*M*P**2*epsilon + 0.000385802469135802*M*P**2 + 0.0339506172839506*M*P*R*epsilon**3 + 0.000771604938271605*M*P*R*epsilon**2 - 0.00694444444444444*M*P*R*epsilon + 0.000578703703703704*M*P*R - 0.0354938271604938*M*P*S*epsilon**3 - 0.00385802469135802*M*P*S*epsilon**2 + 0.00655864197530864*M*P*S*epsilon - 0.000192901234567901*M*P*S - 0.0354938271604938*M*P*T*epsilon**3 - 0.00308641975308642*M*P*T*epsilon**2 + 0.00655864197530864*M*P*T*epsilon - 0.000385802469135802*M*P*T - 0.00308641975308642*M*R*S*epsilon**3 - 0.00385802469135802*M*R*S*epsilon**2 + 0.000578703703703704*M*R*S - 0.0123456790123457*M*R*T*epsilon**3 - 0.00154320987654321*M*R*T*epsilon**2 + 0.00231481481481481*M*R*T*epsilon + 0.00308641975308642*M*S**2*epsilon**3 + 0.00385802469135802*M*S**2*epsilon**2 - 0.000578703703703704*M*S**2 + 0.0138888888888889*M*S*T*epsilon**3 + 0.00462962962962963*M*S*T*epsilon**2 - 0.00192901234567901*M*S*T*epsilon - 0.000385802469135802*M*S*T + 0.00925925925925926*M*T**2*epsilon**3 + 0.00154320987654321*M*T**2*epsilon**2 - 0.00154320987654321*M*T**2*epsilon - 0.0185185185185185*P**3*epsilon**3 - 0.00308641975308642*P**3*epsilon**2 + 0.00308641975308642*P**3*epsilon - 0.0246913580246914*P**2*R*epsilon**3 - 0.00308641975308642*P**2*R*epsilon**2 + 0.00462962962962963*P**2*R*epsilon + 0.0262345679012346*P**2*S*epsilon**3 + 0.00617283950617284*P**2*S*epsilon**2 - 0.00424382716049383*P**2*S*epsilon - 0.000385802469135802*P**2*S + 0.0277777777777778*P**2*T*epsilon**3 + 0.00462962962962963*P**2*T*epsilon**2 - 0.00462962962962963*P**2*T*epsilon + 0.00308641975308642*P*R*S*epsilon**3 + 0.00385802469135802*P*R*S*epsilon**2 - 0.000578703703703704*P*R*S + 0.0123456790123457*P*R*T*epsilon**3 + 0.00154320987654321*P*R*T*epsilon**2 - 0.00231481481481481*P*R*T*epsilon - 0.00308641975308642*P*S**2*epsilon**3 - 0.00385802469135802*P*S**2*epsilon**2 + 0.000578703703703704*P*S**2 - 0.0138888888888889*P*S*T*epsilon**3 - 0.00462962962962963*P*S*T*epsilon**2 + 0.00192901234567901*P*S*T*epsilon + 0.000385802469135802*P*S*T - 0.00925925925925926*P*T**2*epsilon**3 - 0.00154320987654321*P*T**2*epsilon**2 + 0.00154320987654321*P*T**2*epsilon) + N_G*UGA_star*UgA_star*(0.0154320987654321*M**2*P*epsilon**3 - 0.00308641975308642*M**2*P*epsilon**2 - 0.00385802469135802*M**2*P*epsilon + 0.000771604938271605*M**2*P - 0.00925925925925926*M**2*S*epsilon**3 + 0.00231481481481481*M**2*S*epsilon**2 + 0.00231481481481481*M**2*S*epsilon - 0.000578703703703704*M**2*S - 0.00771604938271605*M**2*T*epsilon**3 + 0.00154320987654321*M**2*T*epsilon**2 + 0.00192901234567901*M**2*T*epsilon - 0.000385802469135802*M**2*T - 0.0524691358024691*M*P**2*epsilon**3 - 0.00308641975308642*M*P**2*epsilon**2 + 0.0100308641975309*M*P**2*epsilon - 0.000771604938271605*M*P**2 + 0.037037037037037*M*P*S*epsilon**3 + 0.00694444444444444*M*P*S*epsilon**2 - 0.00617283950617284*M*P*S*epsilon - 0.000192901234567901*M*P*S + 0.0447530864197531*M*P*T*epsilon**3 + 0.00462962962962963*M*P*T*epsilon**2 - 0.00810185185185185*M*P*T*epsilon + 0.000385802469135802*M*P*T - 0.00308641975308642*M*S**2*epsilon**3 - 0.00385802469135802*M*S**2*epsilon**2 + 0.000578703703703704*M*S**2 - 0.0138888888888889*M*S*T*epsilon**3 - 0.00462962962962963*M*S*T*epsilon**2 + 0.00192901234567901*M*S*T*epsilon + 0.000385802469135802*M*S*T - 0.00925925925925926*M*T**2*epsilon**3 - 0.00154320987654321*M*T**2*epsilon**2 + 0.00154320987654321*M*T**2*epsilon + 0.037037037037037*P**3*epsilon**3 + 0.00617283950617284*P**3*epsilon**2 - 0.00617283950617284*P**3*epsilon - 0.0277777777777778*P**2*S*epsilon**3 - 0.00925925925925926*P**2*S*epsilon**2 + 0.00385802469135802*P**2*S*epsilon + 0.000771604938271605*P**2*S - 0.037037037037037*P**2*T*epsilon**3 - 0.00617283950617284*P**2*T*epsilon**2 + 0.00617283950617284*P**2*T*epsilon + 0.00308641975308642*P*S**2*epsilon**3 + 0.00385802469135802*P*S**2*epsilon**2 - 0.000578703703703704*P*S**2 + 0.0138888888888889*P*S*T*epsilon**3 + 0.00462962962962963*P*S*T*epsilon**2 - 0.00192901234567901*P*S*T*epsilon - 0.000385802469135802*P*S*T + 0.00925925925925926*P*T**2*epsilon**3 + 0.00154320987654321*P*T**2*epsilon**2 - 0.00154320987654321*P*T**2*epsilon) + N_G*UGA_star*(-0.00308641975308642*M**3*UGA_prime_star*epsilon**3 + 0.00154320987654321*M**3*UGA_prime_star*epsilon**2 + 0.000771604938271605*M**3*UGA_prime_star*epsilon - 0.000385802469135802*M**3*UGA_prime_star - 0.00308641975308642*M**3*epsilon**3 - 0.000771604938271605*M**3*epsilon**2 + 0.000771604938271605*M**3*epsilon + 0.000192901234567901*M**3 + 0.00925925925925926*M**2*P*UGA_prime_star*epsilon**3 - 0.00154320987654321*M**2*P*UGA_prime_star*epsilon**2 - 0.00231481481481481*M**2*P*UGA_prime_star*epsilon + 0.000385802469135802*M**2*P*UGA_prime_star + 0.0262345679012346*M**2*P*epsilon**3 + 0.0216049382716049*M**2*P*epsilon**2 + 0.00270061728395062*M**2*P*epsilon - 0.000771604938271605*M**2*P - 0.00617283950617284*M**2*R*UGA_prime_star*epsilon**3 + 0.00154320987654321*M**2*R*UGA_prime_star*epsilon + 0.00925925925925926*M**2*S*UGA_prime_star*epsilon**3 - 0.00308641975308642*M**2*S*UGA_prime_star*epsilon**2 - 0.00231481481481481*M**2*S*UGA_prime_star*epsilon + 0.000771604938271605*M**2*S*UGA_prime_star - 0.0108024691358025*M**2*S*epsilon**3 - 0.00848765432098765*M**2*S*epsilon**2 - 0.00192901234567901*M**2*S*epsilon - 0.000192901234567901*M**2*S - 0.00308641975308642*M**2*T*UGA_prime_star*epsilon**3 + 0.000771604938271605*M**2*T*UGA_prime_star*epsilon - 0.0138888888888889*M**2*T*epsilon**3 - 0.00925925925925926*M**2*T*epsilon**2 - 0.00115740740740741*M**2*T*epsilon - 0.00617283950617284*M*P**2*UGA_prime_star*epsilon**3 + 0.00154320987654321*M*P**2*UGA_prime_star*epsilon - 0.0324074074074074*M*P**2*epsilon**3 - 0.0362654320987654*M*P**2*epsilon**2 - 0.00887345679012346*M*P**2*epsilon + 0.000578703703703704*M*P**2 + 0.0123456790123457*M*P*R*UGA_prime_star*epsilon**3 - 0.00308641975308642*M*P*R*UGA_prime_star*epsilon - 0.0216049382716049*M*P*S*UGA_prime_star*epsilon**3 + 0.00308641975308642*M*P*S*UGA_prime_star*epsilon**2 + 0.00540123456790123*M*P*S*UGA_prime_star*epsilon - 0.000771604938271605*M*P*S*UGA_prime_star + 0.0200617283950617*M*P*S*epsilon**3 + 0.0138888888888889*M*P*S*epsilon**2 + 0.00347222222222222*M*P*S*epsilon + 0.000771604938271605*M*P*S + 0.00308641975308642*M*P*T*UGA_prime_star*epsilon**3 - 0.000771604938271605*M*P*T*UGA_prime_star*epsilon + 0.0185185185185185*M*P*T*epsilon**3 + 0.0169753086419753*M*P*T*epsilon**2 + 0.00385802469135802*M*P*T*epsilon + 0.00154320987654321*M*S**2*UGA_prime_star*epsilon**2 - 0.000385802469135802*M*S**2*UGA_prime_star + 0.00308641975308642*M*S*T*UGA_prime_star*epsilon**3 - 0.000771604938271605*M*S*T*UGA_prime_star*epsilon + 0.00925925925925926*P**3*epsilon**3 + 0.0154320987654321*P**3*epsilon**2 + 0.00540123456790123*P**3*epsilon - 0.00617283950617284*P**2*R*UGA_prime_star*epsilon**3 + 0.00154320987654321*P**2*R*UGA_prime_star*epsilon + 0.0123456790123457*P**2*S*UGA_prime_star*epsilon**3 - 0.00308641975308642*P**2*S*UGA_prime_star*epsilon - 0.00925925925925926*P**2*S*epsilon**3 - 0.00540123456790123*P**2*S*epsilon**2 - 0.00154320987654321*P**2*S*epsilon - 0.000578703703703704*P**2*S - 0.00462962962962963*P**2*T*epsilon**3 - 0.00771604938271605*P**2*T*epsilon**2 - 0.00270061728395062*P**2*T*epsilon - 0.00154320987654321*P*S**2*UGA_prime_star*epsilon**2 + 0.000385802469135802*P*S**2*UGA_prime_star - 0.00308641975308642*P*S*T*UGA_prime_star*epsilon**3 + 0.000771604938271605*P*S*T*UGA_prime_star*epsilon) + N_G*UgA_star**2*(0.00462962962962963*M**2*P*epsilon**3 + 0.000771604938271605*M**2*P*epsilon**2 - 0.000771604938271605*M**2*P*epsilon + 0.00617283950617284*M**2*R*epsilon**3 + 0.000771604938271605*M**2*R*epsilon**2 - 0.00115740740740741*M**2*R*epsilon - 0.00617283950617284*M**2*S*epsilon**3 - 0.000771604938271605*M**2*S*epsilon**2 + 0.00115740740740741*M**2*S*epsilon - 0.00462962962962963*M**2*T*epsilon**3 - 0.000771604938271605*M**2*T*epsilon**2 + 0.000771604938271605*M**2*T*epsilon - 0.00925925925925926*M*P**2*epsilon**3 - 0.00154320987654321*M*P**2*epsilon**2 + 0.00154320987654321*M*P**2*epsilon - 0.0123456790123457*M*P*R*epsilon**3 - 0.00154320987654321*M*P*R*epsilon**2 + 0.00231481481481481*M*P*R*epsilon + 0.0123456790123457*M*P*S*epsilon**3 + 0.00154320987654321*M*P*S*epsilon**2 - 0.00231481481481481*M*P*S*epsilon + 0.00925925925925926*M*P*T*epsilon**3 + 0.00154320987654321*M*P*T*epsilon**2 - 0.00154320987654321*M*P*T*epsilon + 0.00462962962962963*P**3*epsilon**3 + 0.000771604938271605*P**3*epsilon**2 - 0.000771604938271605*P**3*epsilon + 0.00617283950617284*P**2*R*epsilon**3 + 0.000771604938271605*P**2*R*epsilon**2 - 0.00115740740740741*P**2*R*epsilon - 0.00617283950617284*P**2*S*epsilon**3 - 0.000771604938271605*P**2*S*epsilon**2 + 0.00115740740740741*P**2*S*epsilon - 0.00462962962962963*P**2*T*epsilon**3 - 0.000771604938271605*P**2*T*epsilon**2 + 0.000771604938271605*P**2*T*epsilon) + N_G*UgA_star*(-0.00925925925925926*M**2*P*epsilon**3 - 0.00154320987654321*M**2*P*epsilon**2 + 0.00154320987654321*M**2*P*epsilon + 0.00617283950617284*M**2*S*epsilon**3 + 0.000771604938271605*M**2*S*epsilon**2 - 0.00115740740740741*M**2*S*epsilon + 0.00462962962962963*M**2*T*epsilon**3 + 0.000771604938271605*M**2*T*epsilon**2 - 0.000771604938271605*M**2*T*epsilon + 0.0185185185185185*M*P**2*epsilon**3 + 0.00308641975308642*M*P**2*epsilon**2 - 0.00308641975308642*M*P**2*epsilon - 0.0123456790123457*M*P*S*epsilon**3 - 0.00154320987654321*M*P*S*epsilon**2 + 0.00231481481481481*M*P*S*epsilon - 0.00925925925925926*M*P*T*epsilon**3 - 0.00154320987654321*M*P*T*epsilon**2 + 0.00154320987654321*M*P*T*epsilon - 0.00925925925925926*P**3*epsilon**3 - 0.00154320987654321*P**3*epsilon**2 + 0.00154320987654321*P**3*epsilon + 0.00617283950617284*P**2*S*epsilon**3 + 0.000771604938271605*P**2*S*epsilon**2 - 0.00115740740740741*P**2*S*epsilon + 0.00462962962962963*P**2*T*epsilon**3 + 0.000771604938271605*P**2*T*epsilon**2 - 0.000771604938271605*P**2*T*epsilon) + N_G*(0.00308641975308642*M**3*UGA_prime_star*epsilon**3 - 0.000771604938271605*M**3*UGA_prime_star*epsilon + 0.00462962962962963*M**3*epsilon**3 + 0.00308641975308642*M**3*epsilon**2 + 0.000385802469135802*M**3*epsilon - 0.00617283950617284*M**2*P*UGA_prime_star*epsilon**3 + 0.00154320987654321*M**2*P*UGA_prime_star*epsilon - 0.00925925925925926*M**2*P*epsilon**3 - 0.00848765432098765*M**2*P*epsilon**2 - 0.00192901234567901*M**2*P*epsilon - 0.00308641975308642*M**2*S*UGA_prime_star*epsilon**3 + 0.000771604938271605*M**2*S*UGA_prime_star*epsilon + 0.00308641975308642*M*P**2*UGA_prime_star*epsilon**3 - 0.000771604938271605*M*P**2*UGA_prime_star*epsilon + 0.00462962962962963*M*P**2*epsilon**3 + 0.00771604938271605*M*P**2*epsilon**2 + 0.00270061728395062*M*P**2*epsilon + 0.00617283950617284*M*P*S*UGA_prime_star*epsilon**3 - 0.00154320987654321*M*P*S*UGA_prime_star*epsilon - 0.00231481481481481*P**3*epsilon**2 - 0.00115740740740741*P**3*epsilon - 0.00308641975308642*P**2*S*UGA_prime_star*epsilon**3 + 0.000771604938271605*P**2*S*UGA_prime_star*epsilon) + UGA_star**8*(0.00617283950617284*P**4*epsilon**3 - 0.00308641975308642*P**4*epsilon**2 + 0.0123456790123457*P**3*R*epsilon**3 - 0.00308641975308642*P**3*R*epsilon - 0.0123456790123457*P**3*S*epsilon**3 + 0.00308641975308642*P**3*S*epsilon - 0.0246913580246914*P**3*T*epsilon**3 + 0.0123456790123457*P**3*T*epsilon**2 + 0.00617283950617284*P**2*R**2*epsilon**3 + 0.00925925925925926*P**2*R**2*epsilon**2 - 0.00462962962962963*P**2*R**2*epsilon - 0.000771604938271605*P**2*R**2 - 0.0123456790123457*P**2*R*S*epsilon**3 - 0.0185185185185185*P**2*R*S*epsilon**2 + 0.00925925925925926*P**2*R*S*epsilon + 0.00154320987654321*P**2*R*S - 0.037037037037037*P**2*R*T*epsilon**3 + 0.00925925925925926*P**2*R*T*epsilon + 0.00617283950617284*P**2*S**2*epsilon**3 + 0.00925925925925926*P**2*S**2*epsilon**2 - 0.00462962962962963*P**2*S**2*epsilon - 0.000771604938271605*P**2*S**2 + 0.037037037037037*P**2*S*T*epsilon**3 - 0.00925925925925926*P**2*S*T*epsilon + 0.037037037037037*P**2*T**2*epsilon**3 - 0.0185185185185185*P**2*T**2*epsilon**2 + 0.00617283950617284*P*R**3*epsilon**2 - 0.00154320987654321*P*R**3 - 0.0185185185185185*P*R**2*S*epsilon**2 + 0.00462962962962963*P*R**2*S - 0.0123456790123457*P*R**2*T*epsilon**3 - 0.0185185185185185*P*R**2*T*epsilon**2 + 0.00925925925925926*P*R**2*T*epsilon + 0.00154320987654321*P*R**2*T + 0.0185185185185185*P*R*S**2*epsilon**2 - 0.00462962962962963*P*R*S**2 + 0.0246913580246914*P*R*S*T*epsilon**3 + 0.037037037037037*P*R*S*T*epsilon**2 - 0.0185185185185185*P*R*S*T*epsilon - 0.00308641975308642*P*R*S*T + 0.037037037037037*P*R*T**2*epsilon**3 - 0.00925925925925926*P*R*T**2*epsilon - 0.00617283950617284*P*S**3*epsilon**2 + 0.00154320987654321*P*S**3 - 0.0123456790123457*P*S**2*T*epsilon**3 - 0.0185185185185185*P*S**2*T*epsilon**2 + 0.00925925925925926*P*S**2*T*epsilon + 0.00154320987654321*P*S**2*T - 0.037037037037037*P*S*T**2*epsilon**3 + 0.00925925925925926*P*S*T**2*epsilon - 0.0246913580246914*P*T**3*epsilon**3 + 0.0123456790123457*P*T**3*epsilon**2 + 0.00154320987654321*R**4*epsilon - 0.000771604938271605*R**4 - 0.00617283950617284*R**3*S*epsilon + 0.00308641975308642*R**3*S - 0.00617283950617284*R**3*T*epsilon**2 + 0.00154320987654321*R**3*T + 0.00925925925925926*R**2*S**2*epsilon - 0.00462962962962963*R**2*S**2 + 0.0185185185185185*R**2*S*T*epsilon**2 - 0.00462962962962963*R**2*S*T + 0.00617283950617284*R**2*T**2*epsilon**3 + 0.00925925925925926*R**2*T**2*epsilon**2 - 0.00462962962962963*R**2*T**2*epsilon - 0.000771604938271605*R**2*T**2 - 0.00617283950617284*R*S**3*epsilon + 0.00308641975308642*R*S**3 - 0.0185185185185185*R*S**2*T*epsilon**2 + 0.00462962962962963*R*S**2*T - 0.0123456790123457*R*S*T**2*epsilon**3 - 0.0185185185185185*R*S*T**2*epsilon**2 + 0.00925925925925926*R*S*T**2*epsilon + 0.00154320987654321*R*S*T**2 - 0.0123456790123457*R*T**3*epsilon**3 + 0.00308641975308642*R*T**3*epsilon + 0.00154320987654321*S**4*epsilon - 0.000771604938271605*S**4 + 0.00617283950617284*S**3*T*epsilon**2 - 0.00154320987654321*S**3*T + 0.00617283950617284*S**2*T**2*epsilon**3 + 0.00925925925925926*S**2*T**2*epsilon**2 - 0.00462962962962963*S**2*T**2*epsilon - 0.000771604938271605*S**2*T**2 + 0.0123456790123457*S*T**3*epsilon**3 - 0.00308641975308642*S*T**3*epsilon + 0.00617283950617284*T**4*epsilon**3 - 0.00308641975308642*T**4*epsilon**2) + UGA_star**7*(0.0123456790123457*M*P**3*epsilon**3 - 0.0123456790123457*M*P**3*epsilon**2 + 0.00308641975308642*M*P**3*epsilon + 0.0246913580246914*M*P**2*R*epsilon**3 - 0.0185185185185185*M*P**2*R*epsilon**2 + 0.00154320987654321*M*P**2*R - 0.0246913580246914*M*P**2*S*epsilon**3 + 0.0185185185185185*M*P**2*S*epsilon**2 - 0.00154320987654321*M*P**2*S - 0.037037037037037*M*P**2*T*epsilon**3 + 0.037037037037037*M*P**2*T*epsilon**2 - 0.00925925925925926*M*P**2*T*epsilon + 0.0123456790123457*M*P*R**2*epsilon**3 - 0.00925925925925926*M*P*R**2*epsilon + 0.00308641975308642*M*P*R**2 - 0.0246913580246914*M*P*R*S*epsilon**3 + 0.0185185185185185*M*P*R*S*epsilon - 0.00617283950617284*M*P*R*S - 0.0493827160493827*M*P*R*T*epsilon**3 + 0.037037037037037*M*P*R*T*epsilon**2 - 0.00308641975308642*M*P*R*T + 0.0123456790123457*M*P*S**2*epsilon**3 - 0.00925925925925926*M*P*S**2*epsilon + 0.00308641975308642*M*P*S**2 + 0.0493827160493827*M*P*S*T*epsilon**3 - 0.037037037037037*M*P*S*T*epsilon**2 + 0.00308641975308642*M*P*S*T + 0.037037037037037*M*P*T**2*epsilon**3 - 0.037037037037037*M*P*T**2*epsilon**2 + 0.00925925925925926*M*P*T**2*epsilon + 0.00617283950617284*M*R**3*epsilon**2 - 0.00617283950617284*M*R**3*epsilon + 0.00154320987654321*M*R**3 - 0.0185185185185185*M*R**2*S*epsilon**2 + 0.0185185185185185*M*R**2*S*epsilon - 0.00462962962962963*M*R**2*S - 0.0123456790123457*M*R**2*T*epsilon**3 + 0.00925925925925926*M*R**2*T*epsilon - 0.00308641975308642*M*R**2*T + 0.0185185185185185*M*R*S**2*epsilon**2 - 0.0185185185185185*M*R*S**2*epsilon + 0.00462962962962963*M*R*S**2 + 0.0246913580246914*M*R*S*T*epsilon**3 - 0.0185185185185185*M*R*S*T*epsilon + 0.00617283950617284*M*R*S*T + 0.0246913580246914*M*R*T**2*epsilon**3 - 0.0185185185185185*M*R*T**2*epsilon**2 + 0.00154320987654321*M*R*T**2 - 0.00617283950617284*M*S**3*epsilon**2 + 0.00617283950617284*M*S**3*epsilon - 0.00154320987654321*M*S**3 - 0.0123456790123457*M*S**2*T*epsilon**3 + 0.00925925925925926*M*S**2*T*epsilon - 0.00308641975308642*M*S**2*T - 0.0246913580246914*M*S*T**2*epsilon**3 + 0.0185185185185185*M*S*T**2*epsilon**2 - 0.00154320987654321*M*S*T**2 - 0.0123456790123457*M*T**3*epsilon**3 + 0.0123456790123457*M*T**3*epsilon**2 - 0.00308641975308642*M*T**3*epsilon - 0.0493827160493827*P**4*epsilon**3 + 0.0246913580246914*P**4*epsilon**2 - 0.0740740740740741*P**3*R*epsilon**3 + 0.0185185185185185*P**3*R*epsilon + 0.0864197530864197*P**3*S*epsilon**3 - 0.0216049382716049*P**3*S*epsilon + 0.172839506172839*P**3*T*epsilon**3 - 0.0864197530864197*P**3*T*epsilon**2 - 0.0246913580246914*P**2*R**2*epsilon**3 - 0.037037037037037*P**2*R**2*epsilon**2 + 0.0185185185185185*P**2*R**2*epsilon + 0.00308641975308642*P**2*R**2 + 0.0617283950617284*P**2*R*S*epsilon**3 + 0.0925925925925926*P**2*R*S*epsilon**2 - 0.0462962962962963*P**2*R*S*epsilon - 0.00771604938271605*P**2*R*S + 0.185185185185185*P**2*R*T*epsilon**3 - 0.0462962962962963*P**2*R*T*epsilon - 0.037037037037037*P**2*S**2*epsilon**3 - 0.0555555555555556*P**2*S**2*epsilon**2 + 0.0277777777777778*P**2*S**2*epsilon + 0.00462962962962963*P**2*S**2 - 0.222222222222222*P**2*S*T*epsilon**3 + 0.0555555555555556*P**2*S*T*epsilon - 0.222222222222222*P**2*T**2*epsilon**3 + 0.111111111111111*P**2*T**2*epsilon**2 - 0.0123456790123457*P*R**3*epsilon**2 + 0.00308641975308642*P*R**3 + 0.0555555555555556*P*R**2*S*epsilon**2 - 0.0138888888888889*P*R**2*S + 0.037037037037037*P*R**2*T*epsilon**3 + 0.0555555555555556*P*R**2*T*epsilon**2 - 0.0277777777777778*P*R**2*T*epsilon - 0.00462962962962963*P*R**2*T - 0.0740740740740741*P*R*S**2*epsilon**2 + 0.0185185185185185*P*R*S**2 - 0.0987654320987654*P*R*S*T*epsilon**3 - 0.148148148148148*P*R*S*T*epsilon**2 + 0.0740740740740741*P*R*S*T*epsilon + 0.0123456790123457*P*R*S*T - 0.148148148148148*P*R*T**2*epsilon**3 + 0.037037037037037*P*R*T**2*epsilon + 0.0308641975308642*P*S**3*epsilon**2 - 0.00771604938271605*P*S**3 + 0.0617283950617284*P*S**2*T*epsilon**3 + 0.0925925925925926*P*S**2*T*epsilon**2 - 0.0462962962962963*P*S**2*T*epsilon - 0.00771604938271605*P*S**2*T + 0.185185185185185*P*S*T**2*epsilon**3 - 0.0462962962962963*P*S*T**2*epsilon + 0.123456790123457*P*T**3*epsilon**3 - 0.0617283950617284*P*T**3*epsilon**2 + 0.00617283950617284*R**3*S*epsilon - 0.00308641975308642*R**3*S + 0.00617283950617284*R**3*T*epsilon**2 - 0.00154320987654321*R**3*T - 0.0185185185185185*R**2*S**2*epsilon + 0.00925925925925926*R**2*S**2 - 0.037037037037037*R**2*S*T*epsilon**2 + 0.00925925925925926*R**2*S*T - 0.0123456790123457*R**2*T**2*epsilon**3 - 0.0185185185185185*R**2*T**2*epsilon**2 + 0.00925925925925926*R**2*T**2*epsilon + 0.00154320987654321*R**2*T**2 + 0.0185185185185185*R*S**3*epsilon - 0.00925925925925926*R*S**3 + 0.0555555555555556*R*S**2*T*epsilon**2 - 0.0138888888888889*R*S**2*T + 0.037037037037037*R*S*T**2*epsilon**3 + 0.0555555555555556*R*S*T**2*epsilon**2 - 0.0277777777777778*R*S*T**2*epsilon - 0.00462962962962963*R*S*T**2 + 0.037037037037037*R*T**3*epsilon**3 - 0.00925925925925926*R*T**3*epsilon - 0.00617283950617284*S**4*epsilon + 0.00308641975308642*S**4 - 0.0246913580246914*S**3*T*epsilon**2 + 0.00617283950617284*S**3*T - 0.0246913580246914*S**2*T**2*epsilon**3 - 0.037037037037037*S**2*T**2*epsilon**2 + 0.0185185185185185*S**2*T**2*epsilon + 0.00308641975308642*S**2*T**2 - 0.0493827160493827*S*T**3*epsilon**3 + 0.0123456790123457*S*T**3*epsilon - 0.0246913580246914*T**4*epsilon**3 + 0.0123456790123457*T**4*epsilon**2) + UGA_star**6*UgA_star**2*(-0.00617283950617284*P**4*epsilon**3 + 0.00308641975308642*P**4*epsilon**2 - 0.0123456790123457*P**3*R*epsilon**3 + 0.00308641975308642*P**3*R*epsilon + 0.0123456790123457*P**3*S*epsilon**3 - 0.00308641975308642*P**3*S*epsilon + 0.0246913580246914*P**3*T*epsilon**3 - 0.0123456790123457*P**3*T*epsilon**2 - 0.00617283950617284*P**2*R**2*epsilon**3 - 0.00925925925925926*P**2*R**2*epsilon**2 + 0.00462962962962963*P**2*R**2*epsilon + 0.000771604938271605*P**2*R**2 + 0.0123456790123457*P**2*R*S*epsilon**3 + 0.0185185185185185*P**2*R*S*epsilon**2 - 0.00925925925925926*P**2*R*S*epsilon - 0.00154320987654321*P**2*R*S + 0.037037037037037*P**2*R*T*epsilon**3 - 0.00925925925925926*P**2*R*T*epsilon - 0.00617283950617284*P**2*S**2*epsilon**3 - 0.00925925925925926*P**2*S**2*epsilon**2 + 0.00462962962962963*P**2*S**2*epsilon + 0.000771604938271605*P**2*S**2 - 0.037037037037037*P**2*S*T*epsilon**3 + 0.00925925925925926*P**2*S*T*epsilon - 0.037037037037037*P**2*T**2*epsilon**3 + 0.0185185185185185*P**2*T**2*epsilon**2 - 0.00617283950617284*P*R**3*epsilon**2 + 0.00154320987654321*P*R**3 + 0.0185185185185185*P*R**2*S*epsilon**2 - 0.00462962962962963*P*R**2*S + 0.0123456790123457*P*R**2*T*epsilon**3 + 0.0185185185185185*P*R**2*T*epsilon**2 - 0.00925925925925926*P*R**2*T*epsilon - 0.00154320987654321*P*R**2*T - 0.0185185185185185*P*R*S**2*epsilon**2 + 0.00462962962962963*P*R*S**2 - 0.0246913580246914*P*R*S*T*epsilon**3 - 0.037037037037037*P*R*S*T*epsilon**2 + 0.0185185185185185*P*R*S*T*epsilon + 0.00308641975308642*P*R*S*T - 0.037037037037037*P*R*T**2*epsilon**3 + 0.00925925925925926*P*R*T**2*epsilon + 0.00617283950617284*P*S**3*epsilon**2 - 0.00154320987654321*P*S**3 + 0.0123456790123457*P*S**2*T*epsilon**3 + 0.0185185185185185*P*S**2*T*epsilon**2 - 0.00925925925925926*P*S**2*T*epsilon - 0.00154320987654321*P*S**2*T + 0.037037037037037*P*S*T**2*epsilon**3 - 0.00925925925925926*P*S*T**2*epsilon + 0.0246913580246914*P*T**3*epsilon**3 - 0.0123456790123457*P*T**3*epsilon**2 - 0.00154320987654321*R**4*epsilon + 0.000771604938271605*R**4 + 0.00617283950617284*R**3*S*epsilon - 0.00308641975308642*R**3*S + 0.00617283950617284*R**3*T*epsilon**2 - 0.00154320987654321*R**3*T - 0.00925925925925926*R**2*S**2*epsilon + 0.00462962962962963*R**2*S**2 - 0.0185185185185185*R**2*S*T*epsilon**2 + 0.00462962962962963*R**2*S*T - 0.00617283950617284*R**2*T**2*epsilon**3 - 0.00925925925925926*R**2*T**2*epsilon**2 + 0.00462962962962963*R**2*T**2*epsilon + 0.000771604938271605*R**2*T**2 + 0.00617283950617284*R*S**3*epsilon - 0.00308641975308642*R*S**3 + 0.0185185185185185*R*S**2*T*epsilon**2 - 0.00462962962962963*R*S**2*T + 0.0123456790123457*R*S*T**2*epsilon**3 + 0.0185185185185185*R*S*T**2*epsilon**2 - 0.00925925925925926*R*S*T**2*epsilon - 0.00154320987654321*R*S*T**2 + 0.0123456790123457*R*T**3*epsilon**3 - 0.00308641975308642*R*T**3*epsilon - 0.00154320987654321*S**4*epsilon + 0.000771604938271605*S**4 - 0.00617283950617284*S**3*T*epsilon**2 + 0.00154320987654321*S**3*T - 0.00617283950617284*S**2*T**2*epsilon**3 - 0.00925925925925926*S**2*T**2*epsilon**2 + 0.00462962962962963*S**2*T**2*epsilon + 0.000771604938271605*S**2*T**2 - 0.0123456790123457*S*T**3*epsilon**3 + 0.00308641975308642*S*T**3*epsilon - 0.00617283950617284*T**4*epsilon**3 + 0.00308641975308642*T**4*epsilon**2) + UGA_star**6*UgA_star*(0.0123456790123457*P**4*epsilon**3 - 0.00617283950617284*P**4*epsilon**2 + 0.0123456790123457*P**3*R*epsilon**3 + 0.00617283950617284*P**3*R*epsilon**2 - 0.00617283950617284*P**3*R*epsilon - 0.0185185185185185*P**3*S*epsilon**3 - 0.00308641975308642*P**3*S*epsilon**2 + 0.00617283950617284*P**3*S*epsilon - 0.0432098765432099*P**3*T*epsilon**3 + 0.0216049382716049*P**3*T*epsilon**2 + 0.0123456790123457*P**2*R**2*epsilon**2 - 0.00308641975308642*P**2*R**2*epsilon - 0.00154320987654321*P**2*R**2 - 0.00617283950617284*P**2*R*S*epsilon**3 - 0.0277777777777778*P**2*R*S*epsilon**2 + 0.00925925925925926*P**2*R*S*epsilon + 0.00308641975308642*P**2*R*S - 0.0308641975308642*P**2*R*T*epsilon**3 - 0.0154320987654321*P**2*R*T*epsilon**2 + 0.0154320987654321*P**2*R*T*epsilon + 0.00617283950617284*P**2*S**2*epsilon**3 + 0.0154320987654321*P**2*S**2*epsilon**2 - 0.00617283950617284*P**2*S**2*epsilon - 0.00154320987654321*P**2*S**2 + 0.0493827160493827*P**2*S*T*epsilon**3 + 0.00617283950617284*P**2*S*T*epsilon**2 - 0.0154320987654321*P**2*S*T*epsilon + 0.0555555555555556*P**2*T**2*epsilon**3 - 0.0277777777777778*P**2*T**2*epsilon**2 + 0.00308641975308642*P*R**3*epsilon - 0.00154320987654321*P*R**3 - 0.00617283950617284*P*R**2*S*epsilon**2 - 0.00771604938271605*P*R**2*S*epsilon + 0.00540123456790123*P*R**2*S - 0.0185185185185185*P*R**2*T*epsilon**2 + 0.00462962962962963*P*R**2*T*epsilon + 0.00231481481481481*P*R**2*T + 0.0123456790123457*P*R*S**2*epsilon**2 + 0.00617283950617284*P*R*S**2*epsilon - 0.00617283950617284*P*R*S**2 + 0.0123456790123457*P*R*S*T*epsilon**3 + 0.0432098765432099*P*R*S*T*epsilon**2 - 0.0154320987654321*P*R*S*T*epsilon - 0.00462962962962963*P*R*S*T + 0.0246913580246914*P*R*T**2*epsilon**3 + 0.0123456790123457*P*R*T**2*epsilon**2 - 0.0123456790123457*P*R*T**2*epsilon - 0.00617283950617284*P*S**3*epsilon**2 - 0.00154320987654321*P*S**3*epsilon + 0.00231481481481481*P*S**3 - 0.0123456790123457*P*S**2*T*epsilon**3 - 0.0246913580246914*P*S**2*T*epsilon**2 + 0.0108024691358025*P*S**2*T*epsilon + 0.00231481481481481*P*S**2*T - 0.0432098765432099*P*S*T**2*epsilon**3 - 0.00308641975308642*P*S*T**2*epsilon**2 + 0.0123456790123457*P*S*T**2*epsilon - 0.0308641975308642*P*T**3*epsilon**3 + 0.0154320987654321*P*T**3*epsilon**2 - 0.00154320987654321*R**3*S*epsilon + 0.000771604938271605*R**3*S - 0.00154320987654321*R**3*T*epsilon + 0.000771604938271605*R**3*T + 0.00462962962962963*R**2*S**2*epsilon - 0.00231481481481481*R**2*S**2 + 0.00617283950617284*R**2*S*T*epsilon**2 + 0.00308641975308642*R**2*S*T*epsilon - 0.00308641975308642*R**2*S*T + 0.00617283950617284*R**2*T**2*epsilon**2 - 0.00154320987654321*R**2*T**2*epsilon - 0.000771604938271605*R**2*T**2 - 0.00462962962962963*R*S**3*epsilon + 0.00231481481481481*R*S**3 - 0.0123456790123457*R*S**2*T*epsilon**2 - 0.00154320987654321*R*S**2*T*epsilon + 0.00385802469135802*R*S**2*T - 0.00617283950617284*R*S*T**2*epsilon**3 - 0.0154320987654321*R*S*T**2*epsilon**2 + 0.00617283950617284*R*S*T**2*epsilon + 0.00154320987654321*R*S*T**2 - 0.00617283950617284*R*T**3*epsilon**3 - 0.00308641975308642*R*T**3*epsilon**2 + 0.00308641975308642*R*T**3*epsilon + 0.00154320987654321*S**4*epsilon - 0.000771604938271605*S**4 + 0.00617283950617284*S**3*T*epsilon**2 - 0.00154320987654321*S**3*T + 0.00617283950617284*S**2*T**2*epsilon**3 + 0.00925925925925926*S**2*T**2*epsilon**2 - 0.00462962962962963*S**2*T**2*epsilon - 0.000771604938271605*S**2*T**2 + 0.0123456790123457*S*T**3*epsilon**3 - 0.00308641975308642*S*T**3*epsilon + 0.00617283950617284*T**4*epsilon**3 - 0.00308641975308642*T**4*epsilon**2) + UGA_star**6*(0.00617283950617284*M**2*P**2*epsilon**3 - 0.00925925925925926*M**2*P**2*epsilon**2 + 0.00462962962962963*M**2*P**2*epsilon - 0.000771604938271605*M**2*P**2 + 0.0123456790123457*M**2*P*R*epsilon**3 - 0.0185185185185185*M**2*P*R*epsilon**2 + 0.00925925925925926*M**2*P*R*epsilon - 0.00154320987654321*M**2*P*R - 0.0123456790123457*M**2*P*S*epsilon**3 + 0.0185185185185185*M**2*P*S*epsilon**2 - 0.00925925925925926*M**2*P*S*epsilon + 0.00154320987654321*M**2*P*S - 0.0123456790123457*M**2*P*T*epsilon**3 + 0.0185185185185185*M**2*P*T*epsilon**2 - 0.00925925925925926*M**2*P*T*epsilon + 0.00154320987654321*M**2*P*T + 0.00617283950617284*M**2*R**2*epsilon**3 - 0.00925925925925926*M**2*R**2*epsilon**2 + 0.00462962962962963*M**2*R**2*epsilon - 0.000771604938271605*M**2*R**2 - 0.0123456790123457*M**2*R*S*epsilon**3 + 0.0185185185185185*M**2*R*S*epsilon**2 - 0.00925925925925926*M**2*R*S*epsilon + 0.00154320987654321*M**2*R*S - 0.0123456790123457*M**2*R*T*epsilon**3 + 0.0185185185185185*M**2*R*T*epsilon**2 - 0.00925925925925926*M**2*R*T*epsilon + 0.00154320987654321*M**2*R*T + 0.00617283950617284*M**2*S**2*epsilon**3 - 0.00925925925925926*M**2*S**2*epsilon**2 + 0.00462962962962963*M**2*S**2*epsilon - 0.000771604938271605*M**2*S**2 + 0.0123456790123457*M**2*S*T*epsilon**3 - 0.0185185185185185*M**2*S*T*epsilon**2 + 0.00925925925925926*M**2*S*T*epsilon - 0.00154320987654321*M**2*S*T + 0.00617283950617284*M**2*T**2*epsilon**3 - 0.00925925925925926*M**2*T**2*epsilon**2 + 0.00462962962962963*M**2*T**2*epsilon - 0.000771604938271605*M**2*T**2 - 0.0987654320987654*M*P**3*epsilon**3 + 0.0864197530864197*M*P**3*epsilon**2 - 0.0185185185185185*M*P**3*epsilon - 0.135802469135802*M*P**2*R*epsilon**3 + 0.0740740740740741*M*P**2*R*epsilon**2 + 0.00925925925925926*M*P**2*R*epsilon - 0.00617283950617284*M*P**2*R + 0.160493827160494*M*P**2*S*epsilon**3 - 0.0925925925925926*M*P**2*S*epsilon**2 - 0.00925925925925926*M*P**2*S*epsilon + 0.00771604938271605*M*P**2*S + 0.259259259259259*M*P**2*T*epsilon**3 - 0.222222222222222*M*P**2*T*epsilon**2 + 0.0462962962962963*M*P**2*T*epsilon - 0.037037037037037*M*P*R**2*epsilon**3 - 0.0185185185185185*M*P*R**2*epsilon**2 + 0.0277777777777778*M*P*R**2*epsilon - 0.00462962962962963*M*P*R**2 + 0.0987654320987654*M*P*R*S*epsilon**3 + 0.037037037037037*M*P*R*S*epsilon**2 - 0.0740740740740741*M*P*R*S*epsilon + 0.0154320987654321*M*P*R*S + 0.222222222222222*M*P*R*T*epsilon**3 - 0.111111111111111*M*P*R*T*epsilon**2 - 0.0185185185185185*M*P*R*T*epsilon + 0.00925925925925926*M*P*R*T - 0.0617283950617284*M*P*S**2*epsilon**3 - 0.0185185185185185*M*P*S**2*epsilon**2 + 0.0462962962962963*M*P*S**2*epsilon - 0.0108024691358025*M*P*S**2 - 0.271604938271605*M*P*S*T*epsilon**3 + 0.148148148148148*M*P*S*T*epsilon**2 + 0.0185185185185185*M*P*S*T*epsilon - 0.0123456790123457*M*P*S*T - 0.222222222222222*M*P*T**2*epsilon**3 + 0.185185185185185*M*P*T**2*epsilon**2 - 0.037037037037037*M*P*T**2*epsilon - 0.00617283950617284*M*R**3*epsilon**2 + 0.00154320987654321*M*R**3 + 0.037037037037037*M*R**2*S*epsilon**2 - 0.0185185185185185*M*R**2*S*epsilon + 0.0246913580246914*M*R**2*T*epsilon**3 + 0.0185185185185185*M*R**2*T*epsilon**2 - 0.0185185185185185*M*R**2*T*epsilon + 0.00154320987654321*M*R**2*T - 0.0555555555555556*M*R*S**2*epsilon**2 + 0.037037037037037*M*R*S**2*epsilon - 0.00462962962962963*M*R*S**2 - 0.0740740740740741*M*R*S*T*epsilon**3 - 0.037037037037037*M*R*S*T*epsilon**2 + 0.0555555555555556*M*R*S*T*epsilon - 0.00925925925925926*M*R*S*T - 0.0864197530864197*M*R*T**2*epsilon**3 + 0.037037037037037*M*R*T**2*epsilon**2 + 0.00925925925925926*M*R*T**2*epsilon - 0.00308641975308642*M*R*T**2 + 0.0246913580246914*M*S**3*epsilon**2 - 0.0185185185185185*M*S**3*epsilon + 0.00308641975308642*M*S**3 + 0.0493827160493827*M*S**2*T*epsilon**3 + 0.0185185185185185*M*S**2*T*epsilon**2 - 0.037037037037037*M*S**2*T*epsilon + 0.00771604938271605*M*S**2*T + 0.111111111111111*M*S*T**2*epsilon**3 - 0.0555555555555556*M*S*T**2*epsilon**2 - 0.00925925925925926*M*S*T**2*epsilon + 0.00462962962962963*M*S*T**2 + 0.0617283950617284*M*T**3*epsilon**3 - 0.0493827160493827*M*T**3*epsilon**2 + 0.00925925925925926*M*T**3*epsilon + 0.166666666666667*P**4*epsilon**3 - 0.0833333333333333*P**4*epsilon**2 + 0.179012345679012*P**3*R*epsilon**3 - 0.00308641975308642*P**3*R*epsilon**2 - 0.0432098765432099*P**3*R*epsilon - 0.253086419753086*P**3*S*epsilon**3 + 0.00308641975308642*P**3*S*epsilon**2 + 0.0617283950617284*P**3*S*epsilon - 0.5*P**3*T*epsilon**3 + 0.25*P**3*T*epsilon**2 + 0.037037037037037*P**2*R**2*epsilon**3 + 0.0493827160493827*P**2*R**2*epsilon**2 - 0.0262345679012346*P**2*R**2*epsilon - 0.00385802469135802*P**2*R**2 - 0.123456790123457*P**2*R*S*epsilon**3 - 0.172839506172839*P**2*R*S*epsilon**2 + 0.0895061728395062*P**2*R*S*epsilon + 0.0138888888888889*P**2*R*S - 0.358024691358025*P**2*R*T*epsilon**3 + 0.00617283950617284*P**2*R*T*epsilon**2 + 0.0864197530864197*P**2*R*T*epsilon + 0.0925925925925926*P**2*S**2*epsilon**3 + 0.132716049382716*P**2*S**2*epsilon**2 - 0.0679012345679012*P**2*S**2*epsilon - 0.0108024691358025*P**2*S**2 + 0.54320987654321*P**2*S*T*epsilon**3 - 0.00617283950617284*P**2*S*T*epsilon**2 - 0.132716049382716*P**2*S*T*epsilon + 0.537037037037037*P**2*T**2*epsilon**3 - 0.268518518518519*P**2*T**2*epsilon**2 + 0.00617283950617284*P*R**3*epsilon**2 - 0.00154320987654321*P*R**3*epsilon - 0.000771604938271605*P*R**3 - 0.0555555555555556*P*R**2*S*epsilon**2 + 0.00462962962962963*P*R**2*S*epsilon + 0.0115740740740741*P*R**2*S - 0.037037037037037*P*R**2*T*epsilon**3 - 0.0493827160493827*P*R**2*T*epsilon**2 + 0.0262345679012346*P*R**2*T*epsilon + 0.00385802469135802*P*R**2*T + 0.111111111111111*P*R*S**2*epsilon**2 - 0.00462962962962963*P*R*S**2*epsilon - 0.025462962962963*P*R*S**2 + 0.148148148148148*P*R*S*T*epsilon**3 + 0.209876543209877*P*R*S*T*epsilon**2 - 0.108024691358025*P*R*S*T*epsilon - 0.0169753086419753*P*R*S*T + 0.216049382716049*P*R*T**2*epsilon**3 - 0.00308641975308642*P*R*T**2*epsilon**2 - 0.0524691358024691*P*R*T**2*epsilon - 0.0617283950617284*P*S**3*epsilon**2 + 0.00154320987654321*P*S**3*epsilon + 0.0146604938271605*P*S**3 - 0.123456790123457*P*S**2*T*epsilon**3 - 0.179012345679012*P*S**2*T*epsilon**2 + 0.0910493827160494*P*S**2*T*epsilon + 0.0146604938271605*P*S**2*T - 0.364197530864198*P*S*T**2*epsilon**3 + 0.00308641975308642*P*S*T**2*epsilon**2 + 0.0895061728395062*P*S*T**2*epsilon - 0.240740740740741*P*T**3*epsilon**3 + 0.12037037037037*P*T**3*epsilon**2 + 0.00925925925925926*R**2*S**2*epsilon - 0.00462962962962963*R**2*S**2 + 0.0185185185185185*R**2*S*T*epsilon**2 - 0.00462962962962963*R**2*S*T + 0.00617283950617284*R**2*T**2*epsilon**3 + 0.00925925925925926*R**2*T**2*epsilon**2 - 0.00462962962962963*R**2*T**2*epsilon - 0.000771604938271605*R**2*T**2 - 0.0185185185185185*R*S**3*epsilon + 0.00925925925925926*R*S**3 - 0.0555555555555556*R*S**2*T*epsilon**2 + 0.0138888888888889*R*S**2*T - 0.037037037037037*R*S*T**2*epsilon**3 - 0.0555555555555556*R*S*T**2*epsilon**2 + 0.0277777777777778*R*S*T**2*epsilon + 0.00462962962962963*R*S*T**2 - 0.037037037037037*R*T**3*epsilon**3 + 0.00925925925925926*R*T**3*epsilon + 0.00925925925925926*S**4*epsilon - 0.00462962962962963*S**4 + 0.037037037037037*S**3*T*epsilon**2 - 0.00925925925925926*S**3*T + 0.037037037037037*S**2*T**2*epsilon**3 + 0.0555555555555556*S**2*T**2*epsilon**2 - 0.0277777777777778*S**2*T**2*epsilon - 0.00462962962962963*S**2*T**2 + 0.0740740740740741*S*T**3*epsilon**3 - 0.0185185185185185*S*T**3*epsilon + 0.037037037037037*T**4*epsilon**3 - 0.0185185185185185*T**4*epsilon**2) + UGA_star**5*UgA_star**2*(-0.0123456790123457*M*P**3*epsilon**3 + 0.0123456790123457*M*P**3*epsilon**2 - 0.00308641975308642*M*P**3*epsilon - 0.0246913580246914*M*P**2*R*epsilon**3 + 0.0185185185185185*M*P**2*R*epsilon**2 - 0.00154320987654321*M*P**2*R + 0.0246913580246914*M*P**2*S*epsilon**3 - 0.0185185185185185*M*P**2*S*epsilon**2 + 0.00154320987654321*M*P**2*S + 0.037037037037037*M*P**2*T*epsilon**3 - 0.037037037037037*M*P**2*T*epsilon**2 + 0.00925925925925926*M*P**2*T*epsilon - 0.0123456790123457*M*P*R**2*epsilon**3 + 0.00925925925925926*M*P*R**2*epsilon - 0.00308641975308642*M*P*R**2 + 0.0246913580246914*M*P*R*S*epsilon**3 - 0.0185185185185185*M*P*R*S*epsilon + 0.00617283950617284*M*P*R*S + 0.0493827160493827*M*P*R*T*epsilon**3 - 0.037037037037037*M*P*R*T*epsilon**2 + 0.00308641975308642*M*P*R*T - 0.0123456790123457*M*P*S**2*epsilon**3 + 0.00925925925925926*M*P*S**2*epsilon - 0.00308641975308642*M*P*S**2 - 0.0493827160493827*M*P*S*T*epsilon**3 + 0.037037037037037*M*P*S*T*epsilon**2 - 0.00308641975308642*M*P*S*T - 0.037037037037037*M*P*T**2*epsilon**3 + 0.037037037037037*M*P*T**2*epsilon**2 - 0.00925925925925926*M*P*T**2*epsilon - 0.00617283950617284*M*R**3*epsilon**2 + 0.00617283950617284*M*R**3*epsilon - 0.00154320987654321*M*R**3 + 0.0185185185185185*M*R**2*S*epsilon**2 - 0.0185185185185185*M*R**2*S*epsilon + 0.00462962962962963*M*R**2*S + 0.0123456790123457*M*R**2*T*epsilon**3 - 0.00925925925925926*M*R**2*T*epsilon + 0.00308641975308642*M*R**2*T - 0.0185185185185185*M*R*S**2*epsilon**2 + 0.0185185185185185*M*R*S**2*epsilon - 0.00462962962962963*M*R*S**2 - 0.0246913580246914*M*R*S*T*epsilon**3 + 0.0185185185185185*M*R*S*T*epsilon - 0.00617283950617284*M*R*S*T - 0.0246913580246914*M*R*T**2*epsilon**3 + 0.0185185185185185*M*R*T**2*epsilon**2 - 0.00154320987654321*M*R*T**2 + 0.00617283950617284*M*S**3*epsilon**2 - 0.00617283950617284*M*S**3*epsilon + 0.00154320987654321*M*S**3 + 0.0123456790123457*M*S**2*T*epsilon**3 - 0.00925925925925926*M*S**2*T*epsilon + 0.00308641975308642*M*S**2*T + 0.0246913580246914*M*S*T**2*epsilon**3 - 0.0185185185185185*M*S*T**2*epsilon**2 + 0.00154320987654321*M*S*T**2 + 0.0123456790123457*M*T**3*epsilon**3 - 0.0123456790123457*M*T**3*epsilon**2 + 0.00308641975308642*M*T**3*epsilon + 0.037037037037037*P**4*epsilon**3 - 0.0185185185185185*P**4*epsilon**2 + 0.0617283950617284*P**3*R*epsilon**3 - 0.00617283950617284*P**3*R*epsilon**2 - 0.0123456790123457*P**3*R*epsilon - 0.0679012345679012*P**3*S*epsilon**3 + 0.00308641975308642*P**3*S*epsilon**2 + 0.0154320987654321*P**3*S*epsilon - 0.12962962962963*P**3*T*epsilon**3 + 0.0648148148148148*P**3*T*epsilon**2 + 0.0246913580246914*P**2*R**2*epsilon**3 + 0.0246913580246914*P**2*R**2*epsilon**2 - 0.0154320987654321*P**2*R**2*epsilon - 0.00154320987654321*P**2*R**2 - 0.0555555555555556*P**2*R*S*epsilon**3 - 0.0648148148148148*P**2*R*S*epsilon**2 + 0.037037037037037*P**2*R*S*epsilon + 0.00462962962962963*P**2*R*S - 0.154320987654321*P**2*R*T*epsilon**3 + 0.0154320987654321*P**2*R*T*epsilon**2 + 0.0308641975308642*P**2*R*T*epsilon + 0.0308641975308642*P**2*S**2*epsilon**3 + 0.0401234567901235*P**2*S**2*epsilon**2 - 0.0216049382716049*P**2*S**2*epsilon - 0.00308641975308642*P**2*S**2 + 0.172839506172839*P**2*S*T*epsilon**3 - 0.00617283950617284*P**2*S*T*epsilon**2 - 0.0401234567901235*P**2*S*T*epsilon + 0.166666666666667*P**2*T**2*epsilon**3 - 0.0833333333333333*P**2*T**2*epsilon**2 + 0.0123456790123457*P*R**3*epsilon**2 - 0.00308641975308642*P*R**3*epsilon - 0.00154320987654321*P*R**3 - 0.0493827160493827*P*R**2*S*epsilon**2 + 0.00771604938271605*P*R**2*S*epsilon + 0.00848765432098765*P*R**2*S - 0.037037037037037*P*R**2*T*epsilon**3 - 0.037037037037037*P*R**2*T*epsilon**2 + 0.0231481481481481*P*R**2*T*epsilon + 0.00231481481481481*P*R**2*T + 0.0617283950617284*P*R*S**2*epsilon**2 - 0.00617283950617284*P*R*S**2*epsilon - 0.0123456790123457*P*R*S**2 + 0.0864197530864197*P*R*S*T*epsilon**3 + 0.104938271604938*P*R*S*T*epsilon**2 - 0.058641975308642*P*R*S*T*epsilon - 0.00771604938271605*P*R*S*T + 0.123456790123457*P*R*T**2*epsilon**3 - 0.0123456790123457*P*R*T**2*epsilon**2 - 0.0246913580246914*P*R*T**2*epsilon - 0.0246913580246914*P*S**3*epsilon**2 + 0.00154320987654321*P*S**3*epsilon + 0.00540123456790123*P*S**3 - 0.0493827160493827*P*S**2*T*epsilon**3 - 0.0679012345679012*P*S**2*T*epsilon**2 + 0.0354938271604938*P*S**2*T*epsilon + 0.00540123456790123*P*S**2*T - 0.141975308641975*P*S*T**2*epsilon**3 + 0.00308641975308642*P*S*T**2*epsilon**2 + 0.0339506172839506*P*S*T**2*epsilon - 0.0925925925925926*P*T**3*epsilon**3 + 0.0462962962962963*P*T**3*epsilon**2 - 0.00462962962962963*R**3*S*epsilon + 0.00231481481481481*R**3*S - 0.00617283950617284*R**3*T*epsilon**2 + 0.00154320987654321*R**3*T*epsilon + 0.000771604938271605*R**3*T + 0.0138888888888889*R**2*S**2*epsilon - 0.00694444444444444*R**2*S**2 + 0.0308641975308642*R**2*S*T*epsilon**2 - 0.00308641975308642*R**2*S*T*epsilon - 0.00617283950617284*R**2*S*T + 0.0123456790123457*R**2*T**2*epsilon**3 + 0.0123456790123457*R**2*T**2*epsilon**2 - 0.00771604938271605*R**2*T**2*epsilon - 0.000771604938271605*R**2*T**2 - 0.0138888888888889*R*S**3*epsilon + 0.00694444444444444*R*S**3 - 0.0432098765432099*R*S**2*T*epsilon**2 + 0.00154320987654321*R*S**2*T*epsilon + 0.0100308641975309*R*S**2*T - 0.0308641975308642*R*S*T**2*epsilon**3 - 0.0401234567901235*R*S*T**2*epsilon**2 + 0.0216049382716049*R*S*T**2*epsilon + 0.00308641975308642*R*S*T**2 - 0.0308641975308642*R*T**3*epsilon**3 + 0.00308641975308642*R*T**3*epsilon**2 + 0.00617283950617284*R*T**3*epsilon + 0.00462962962962963*S**4*epsilon - 0.00231481481481481*S**4 + 0.0185185185185185*S**3*T*epsilon**2 - 0.00462962962962963*S**3*T + 0.0185185185185185*S**2*T**2*epsilon**3 + 0.0277777777777778*S**2*T**2*epsilon**2 - 0.0138888888888889*S**2*T**2*epsilon - 0.00231481481481481*S**2*T**2 + 0.037037037037037*S*T**3*epsilon**3 - 0.00925925925925926*S*T**3*epsilon + 0.0185185185185185*T**4*epsilon**3 - 0.00925925925925926*T**4*epsilon**2) + UGA_star**5*UgA_star*(0.0246913580246914*M*P**3*epsilon**3 - 0.0246913580246914*M*P**3*epsilon**2 + 0.00617283950617284*M*P**3*epsilon + 0.0246913580246914*M*P**2*R*epsilon**3 - 0.0123456790123457*M*P**2*R*epsilon**2 - 0.00617283950617284*M*P**2*R*epsilon + 0.00308641975308642*M*P**2*R - 0.037037037037037*M*P**2*S*epsilon**3 + 0.0246913580246914*M*P**2*S*epsilon**2 + 0.00308641975308642*M*P**2*S*epsilon - 0.00308641975308642*M*P**2*S - 0.0617283950617284*M*P**2*T*epsilon**3 + 0.0617283950617284*M*P**2*T*epsilon**2 - 0.0154320987654321*M*P**2*T*epsilon + 0.0123456790123457*M*P*R**2*epsilon**2 - 0.0123456790123457*M*P*R**2*epsilon + 0.00308641975308642*M*P*R**2 - 0.0123456790123457*M*P*R*S*epsilon**3 - 0.0185185185185185*M*P*R*S*epsilon**2 + 0.0277777777777778*M*P*R*S*epsilon - 0.00771604938271605*M*P*R*S - 0.037037037037037*M*P*R*T*epsilon**3 + 0.0185185185185185*M*P*R*T*epsilon**2 + 0.00925925925925926*M*P*R*T*epsilon - 0.00462962962962963*M*P*R*T + 0.0123456790123457*M*P*S**2*epsilon**3 + 0.00617283950617284*M*P*S**2*epsilon**2 - 0.0154320987654321*M*P*S**2*epsilon + 0.00462962962962963*M*P*S**2 + 0.0617283950617284*M*P*S*T*epsilon**3 - 0.0432098765432099*M*P*S*T*epsilon**2 - 0.00308641975308642*M*P*S*T*epsilon + 0.00462962962962963*M*P*S*T + 0.0493827160493827*M*P*T**2*epsilon**3 - 0.0493827160493827*M*P*T**2*epsilon**2 + 0.0123456790123457*M*P*T**2*epsilon - 0.00617283950617284*M*R**2*S*epsilon**2 + 0.00617283950617284*M*R**2*S*epsilon - 0.00154320987654321*M*R**2*S - 0.00617283950617284*M*R**2*T*epsilon**2 + 0.00617283950617284*M*R**2*T*epsilon - 0.00154320987654321*M*R**2*T + 0.0123456790123457*M*R*S**2*epsilon**2 - 0.0123456790123457*M*R*S**2*epsilon + 0.00308641975308642*M*R*S**2 + 0.0123456790123457*M*R*S*T*epsilon**3 + 0.00617283950617284*M*R*S*T*epsilon**2 - 0.0154320987654321*M*R*S*T*epsilon + 0.00462962962962963*M*R*S*T + 0.0123456790123457*M*R*T**2*epsilon**3 - 0.00617283950617284*M*R*T**2*epsilon**2 - 0.00308641975308642*M*R*T**2*epsilon + 0.00154320987654321*M*R*T**2 - 0.00617283950617284*M*S**3*epsilon**2 + 0.00617283950617284*M*S**3*epsilon - 0.00154320987654321*M*S**3 - 0.0123456790123457*M*S**2*T*epsilon**3 + 0.00925925925925926*M*S**2*T*epsilon - 0.00308641975308642*M*S**2*T - 0.0246913580246914*M*S*T**2*epsilon**3 + 0.0185185185185185*M*S*T**2*epsilon**2 - 0.00154320987654321*M*S*T**2 - 0.0123456790123457*M*T**3*epsilon**3 + 0.0123456790123457*M*T**3*epsilon**2 - 0.00308641975308642*M*T**3*epsilon - 0.0740740740740741*P**4*epsilon**3 + 0.037037037037037*P**4*epsilon**2 - 0.0493827160493827*P**3*R*epsilon**3 - 0.0246913580246914*P**3*R*epsilon**2 + 0.0246913580246914*P**3*R*epsilon + 0.0987654320987654*P**3*S*epsilon**3 + 0.0123456790123457*P**3*S*epsilon**2 - 0.0308641975308642*P**3*S*epsilon + 0.222222222222222*P**3*T*epsilon**3 - 0.111111111111111*P**3*T*epsilon**2 - 0.0246913580246914*P**2*R**2*epsilon**2 + 0.00617283950617284*P**2*R**2*epsilon + 0.00308641975308642*P**2*R**2 + 0.0246913580246914*P**2*R*S*epsilon**3 + 0.0864197530864197*P**2*R*S*epsilon**2 - 0.0308641975308642*P**2*R*S*epsilon - 0.00925925925925926*P**2*R*S + 0.0987654320987654*P**2*R*T*epsilon**3 + 0.0493827160493827*P**2*R*T*epsilon**2 - 0.0493827160493827*P**2*R*T*epsilon - 0.0308641975308642*P**2*S**2*epsilon**3 - 0.0648148148148148*P**2*S**2*epsilon**2 + 0.0277777777777778*P**2*S**2*epsilon + 0.00617283950617284*P**2*S**2 - 0.222222222222222*P**2*S*T*epsilon**3 - 0.0185185185185185*P**2*S*T*epsilon**2 + 0.0648148148148148*P**2*S*T*epsilon - 0.240740740740741*P**2*T**2*epsilon**3 + 0.12037037037037*P**2*T**2*epsilon**2 + 0.0123456790123457*P*R**2*S*epsilon**2 + 0.00617283950617284*P*R**2*S*epsilon - 0.00617283950617284*P*R**2*S + 0.0246913580246914*P*R**2*T*epsilon**2 - 0.00617283950617284*P*R**2*T*epsilon - 0.00308641975308642*P*R**2*T - 0.037037037037037*P*R*S**2*epsilon**2 - 0.00925925925925926*P*R*S**2*epsilon + 0.0138888888888889*P*R*S**2 - 0.037037037037037*P*R*S*T*epsilon**3 - 0.104938271604938*P*R*S*T*epsilon**2 + 0.0401234567901235*P*R*S*T*epsilon + 0.0108024691358025*P*R*S*T - 0.0617283950617284*P*R*T**2*epsilon**3 - 0.0308641975308642*P*R*T**2*epsilon**2 + 0.0308641975308642*P*R*T**2*epsilon + 0.0246913580246914*P*S**3*epsilon**2 + 0.00308641975308642*P*S**3*epsilon - 0.00771604938271605*P*S**3 + 0.0493827160493827*P*S**2*T*epsilon**3 + 0.0864197530864197*P*S**2*T*epsilon**2 - 0.0401234567901235*P*S**2*T*epsilon - 0.00771604938271605*P*S**2*T + 0.160493827160494*P*S*T**2*epsilon**3 + 0.00617283950617284*P*S*T**2*epsilon**2 - 0.0432098765432099*P*S*T**2*epsilon + 0.111111111111111*P*T**3*epsilon**3 - 0.0555555555555556*P*T**3*epsilon**2 - 0.00462962962962963*R**2*S**2*epsilon + 0.00231481481481481*R**2*S**2 - 0.00617283950617284*R**2*S*T*epsilon**2 - 0.00308641975308642*R**2*S*T*epsilon + 0.00308641975308642*R**2*S*T - 0.00617283950617284*R**2*T**2*epsilon**2 + 0.00154320987654321*R**2*T**2*epsilon + 0.000771604938271605*R**2*T**2 + 0.00925925925925926*R*S**3*epsilon - 0.00462962962962963*R*S**3 + 0.0246913580246914*R*S**2*T*epsilon**2 + 0.00308641975308642*R*S**2*T*epsilon - 0.00771604938271605*R*S**2*T + 0.0123456790123457*R*S*T**2*epsilon**3 + 0.0308641975308642*R*S*T**2*epsilon**2 - 0.0123456790123457*R*S*T**2*epsilon - 0.00308641975308642*R*S*T**2 + 0.0123456790123457*R*T**3*epsilon**3 + 0.00617283950617284*R*T**3*epsilon**2 - 0.00617283950617284*R*T**3*epsilon - 0.00462962962962963*S**4*epsilon + 0.00231481481481481*S**4 - 0.0185185185185185*S**3*T*epsilon**2 + 0.00462962962962963*S**3*T - 0.0185185185185185*S**2*T**2*epsilon**3 - 0.0277777777777778*S**2*T**2*epsilon**2 + 0.0138888888888889*S**2*T**2*epsilon + 0.00231481481481481*S**2*T**2 - 0.037037037037037*S*T**3*epsilon**3 + 0.00925925925925926*S*T**3*epsilon - 0.0185185185185185*T**4*epsilon**3 + 0.00925925925925926*T**4*epsilon**2) + UGA_star**5*(-0.0617283950617284*M**2*P**2*epsilon**3 + 0.0740740740740741*M**2*P**2*epsilon**2 - 0.0277777777777778*M**2*P**2*epsilon + 0.00308641975308642*M**2*P**2 - 0.0740740740740741*M**2*P*R*epsilon**3 + 0.0740740740740741*M**2*P*R*epsilon**2 - 0.0185185185185185*M**2*P*R*epsilon + 0.0864197530864197*M**2*P*S*epsilon**3 - 0.0925925925925926*M**2*P*S*epsilon**2 + 0.0277777777777778*M**2*P*S*epsilon - 0.00154320987654321*M**2*P*S + 0.111111111111111*M**2*P*T*epsilon**3 - 0.12962962962963*M**2*P*T*epsilon**2 + 0.0462962962962963*M**2*P*T*epsilon - 0.00462962962962963*M**2*P*T - 0.0123456790123457*M**2*R**2*epsilon**3 + 0.00925925925925926*M**2*R**2*epsilon - 0.00308641975308642*M**2*R**2 + 0.037037037037037*M**2*R*S*epsilon**3 - 0.0185185185185185*M**2*R*S*epsilon**2 - 0.00925925925925926*M**2*R*S*epsilon + 0.00462962962962963*M**2*R*S + 0.0617283950617284*M**2*R*T*epsilon**3 - 0.0555555555555556*M**2*R*T*epsilon**2 + 0.00925925925925926*M**2*R*T*epsilon + 0.00154320987654321*M**2*R*T - 0.0246913580246914*M**2*S**2*epsilon**3 + 0.0185185185185185*M**2*S**2*epsilon**2 - 0.00154320987654321*M**2*S**2 - 0.0740740740740741*M**2*S*T*epsilon**3 + 0.0740740740740741*M**2*S*T*epsilon**2 - 0.0185185185185185*M**2*S*T*epsilon - 0.0493827160493827*M**2*T**2*epsilon**3 + 0.0555555555555556*M**2*T**2*epsilon**2 - 0.0185185185185185*M**2*T**2*epsilon + 0.00154320987654321*M**2*T**2 + 0.320987654320988*M*P**3*epsilon**3 - 0.246913580246914*M*P**3*epsilon**2 + 0.0432098765432099*M*P**3*epsilon + 0.283950617283951*M*P**2*R*epsilon**3 - 0.104938271604938*M*P**2*R*epsilon**2 - 0.0339506172839506*M*P**2*R*epsilon + 0.00771604938271605*M*P**2*R - 0.419753086419753*M*P**2*S*epsilon**3 + 0.179012345679012*M*P**2*S*epsilon**2 + 0.0432098765432099*M*P**2*S*epsilon - 0.0138888888888889*M*P**2*S - 0.716049382716049*M*P**2*T*epsilon**3 + 0.530864197530864*M*P**2*T*epsilon**2 - 0.0864197530864197*M*P**2*T*epsilon + 0.037037037037037*M*P*R**2*epsilon**3 + 0.0308641975308642*M*P*R**2*epsilon**2 - 0.0216049382716049*M*P*R**2*epsilon - 0.00154320987654321*M*P*R**2 - 0.148148148148148*M*P*R*S*epsilon**3 - 0.0987654320987654*M*P*R*S*epsilon**2 + 0.0987654320987654*M*P*R*S*epsilon - 0.00617283950617284*M*P*R*S - 0.358024691358025*M*P*R*T*epsilon**3 + 0.104938271604938*M*P*R*T*epsilon**2 + 0.0524691358024691*M*P*R*T*epsilon - 0.00771604938271605*M*P*R*T + 0.123456790123457*M*P*S**2*epsilon**3 + 0.0679012345679012*M*P*S**2*epsilon**2 - 0.0864197530864197*M*P*S**2*epsilon + 0.0108024691358025*M*P*S**2 + 0.580246913580247*M*P*S*T*epsilon**3 - 0.216049382716049*M*P*S*T*epsilon**2 - 0.0709876543209877*M*P*S*T*epsilon + 0.0169753086419753*M*P*S*T + 0.506172839506173*M*P*T**2*epsilon**3 - 0.358024691358025*M*P*T**2*epsilon**2 + 0.0524691358024691*M*P*T**2*epsilon - 0.0185185185185185*M*R**2*S*epsilon**2 + 0.00462962962962963*M*R**2*S - 0.0123456790123457*M*R**2*T*epsilon**3 - 0.0185185185185185*M*R**2*T*epsilon**2 + 0.00925925925925926*M*R**2*T*epsilon + 0.00154320987654321*M*R**2*T + 0.0555555555555556*M*R*S**2*epsilon**2 - 0.0185185185185185*M*R*S**2*epsilon - 0.00462962962962963*M*R*S**2 + 0.0740740740740741*M*R*S*T*epsilon**3 + 0.0740740740740741*M*R*S*T*epsilon**2 - 0.0555555555555556*M*R*S*T*epsilon + 0.0987654320987654*M*R*T**2*epsilon**3 - 0.0185185185185185*M*R*T**2*epsilon**2 - 0.0185185185185185*M*R*T**2*epsilon + 0.00154320987654321*M*R*T**2 - 0.037037037037037*M*S**3*epsilon**2 + 0.0185185185185185*M*S**3*epsilon - 0.0740740740740741*M*S**2*T*epsilon**3 - 0.0555555555555556*M*S**2*T*epsilon**2 + 0.0555555555555556*M*S**2*T*epsilon - 0.00462962962962963*M*S**2*T - 0.185185185185185*M*S*T**2*epsilon**3 + 0.0555555555555556*M*S*T**2*epsilon**2 + 0.0277777777777778*M*S*T**2*epsilon - 0.00462962962962963*M*S*T**2 - 0.111111111111111*M*T**3*epsilon**3 + 0.0740740740740741*M*T**3*epsilon**2 - 0.00925925925925926*M*T**3*epsilon - 0.308641975308642*P**4*epsilon**3 + 0.154320987654321*P**4*epsilon**2 - 0.222222222222222*P**3*R*epsilon**3 + 0.0123456790123457*P**3*R*epsilon**2 + 0.0493827160493827*P**3*R*epsilon + 0.401234567901235*P**3*S*epsilon**3 - 0.0154320987654321*P**3*S*epsilon**2 - 0.0925925925925926*P**3*S*epsilon + 0.771604938271605*P**3*T*epsilon**3 - 0.385802469135802*P**3*T*epsilon**2 - 0.0246913580246914*P**2*R**2*epsilon**3 - 0.0246913580246914*P**2*R**2*epsilon**2 + 0.0154320987654321*P**2*R**2*epsilon + 0.00154320987654321*P**2*R**2 + 0.123456790123457*P**2*R*S*epsilon**3 + 0.148148148148148*P**2*R*S*epsilon**2 - 0.0833333333333333*P**2*R*S*epsilon - 0.0108024691358025*P**2*R*S + 0.333333333333333*P**2*R*T*epsilon**3 - 0.0185185185185185*P**2*R*T*epsilon**2 - 0.0740740740740741*P**2*R*T*epsilon - 0.123456790123457*P**2*S**2*epsilon**3 - 0.160493827160494*P**2*S**2*epsilon**2 + 0.0864197530864197*P**2*S**2*epsilon + 0.0123456790123457*P**2*S**2 - 0.691358024691358*P**2*S*T*epsilon**3 + 0.0246913580246914*P**2*S*T*epsilon**2 + 0.160493827160494*P**2*S*T*epsilon - 0.666666666666667*P**2*T**2*epsilon**3 + 0.333333333333333*P**2*T**2*epsilon**2 + 0.0185185185185185*P*R**2*S*epsilon**2 - 0.00462962962962963*P*R**2*S*epsilon - 0.00231481481481481*P*R**2*S + 0.0123456790123457*P*R**2*T*epsilon**3 + 0.0123456790123457*P*R**2*T*epsilon**2 - 0.00771604938271605*P*R**2*T*epsilon - 0.000771604938271605*P*R**2*T - 0.0740740740740741*P*R*S**2*epsilon**2 + 0.00925925925925926*P*R*S**2*epsilon + 0.0138888888888889*P*R*S**2 - 0.0987654320987654*P*R*S*T*epsilon**3 - 0.123456790123457*P*R*S*T*epsilon**2 + 0.0679012345679012*P*R*S*T*epsilon + 0.00925925925925926*P*R*S*T - 0.135802469135802*P*R*T**2*epsilon**3 + 0.00617283950617284*P*R*T**2*epsilon**2 + 0.0308641975308642*P*R*T**2*epsilon + 0.0617283950617284*P*S**3*epsilon**2 - 0.00462962962962963*P*S**3*epsilon - 0.0131172839506173*P*S**3 + 0.123456790123457*P*S**2*T*epsilon**3 + 0.166666666666667*P*S**2*T*epsilon**2 - 0.087962962962963*P*S**2*T*epsilon - 0.0131172839506173*P*S**2*T + 0.351851851851852*P*S*T**2*epsilon**3 - 0.00925925925925926*P*S*T**2*epsilon**2 - 0.0833333333333333*P*S*T**2*epsilon + 0.228395061728395*P*T**3*epsilon**3 - 0.114197530864198*P*T**3*epsilon**2 + 0.00617283950617284*R*S**3*epsilon - 0.00308641975308642*R*S**3 + 0.0185185185185185*R*S**2*T*epsilon**2 - 0.00462962962962963*R*S**2*T + 0.0123456790123457*R*S*T**2*epsilon**3 + 0.0185185185185185*R*S*T**2*epsilon**2 - 0.00925925925925926*R*S*T**2*epsilon - 0.00154320987654321*R*S*T**2 + 0.0123456790123457*R*T**3*epsilon**3 - 0.00308641975308642*R*T**3*epsilon - 0.00617283950617284*S**4*epsilon + 0.00308641975308642*S**4 - 0.0246913580246914*S**3*T*epsilon**2 + 0.00617283950617284*S**3*T - 0.0246913580246914*S**2*T**2*epsilon**3 - 0.037037037037037*S**2*T**2*epsilon**2 + 0.0185185185185185*S**2*T**2*epsilon + 0.00308641975308642*S**2*T**2 - 0.0493827160493827*S*T**3*epsilon**3 + 0.0123456790123457*S*T**3*epsilon - 0.0246913580246914*T**4*epsilon**3 + 0.0123456790123457*T**4*epsilon**2) + UGA_star**4*UgA_star**2*(-0.00617283950617284*M**2*P**2*epsilon**3 + 0.00925925925925926*M**2*P**2*epsilon**2 - 0.00462962962962963*M**2*P**2*epsilon + 0.000771604938271605*M**2*P**2 - 0.0123456790123457*M**2*P*R*epsilon**3 + 0.0185185185185185*M**2*P*R*epsilon**2 - 0.00925925925925926*M**2*P*R*epsilon + 0.00154320987654321*M**2*P*R + 0.0123456790123457*M**2*P*S*epsilon**3 - 0.0185185185185185*M**2*P*S*epsilon**2 + 0.00925925925925926*M**2*P*S*epsilon - 0.00154320987654321*M**2*P*S + 0.0123456790123457*M**2*P*T*epsilon**3 - 0.0185185185185185*M**2*P*T*epsilon**2 + 0.00925925925925926*M**2*P*T*epsilon - 0.00154320987654321*M**2*P*T - 0.00617283950617284*M**2*R**2*epsilon**3 + 0.00925925925925926*M**2*R**2*epsilon**2 - 0.00462962962962963*M**2*R**2*epsilon + 0.000771604938271605*M**2*R**2 + 0.0123456790123457*M**2*R*S*epsilon**3 - 0.0185185185185185*M**2*R*S*epsilon**2 + 0.00925925925925926*M**2*R*S*epsilon - 0.00154320987654321*M**2*R*S + 0.0123456790123457*M**2*R*T*epsilon**3 - 0.0185185185185185*M**2*R*T*epsilon**2 + 0.00925925925925926*M**2*R*T*epsilon - 0.00154320987654321*M**2*R*T - 0.00617283950617284*M**2*S**2*epsilon**3 + 0.00925925925925926*M**2*S**2*epsilon**2 - 0.00462962962962963*M**2*S**2*epsilon + 0.000771604938271605*M**2*S**2 - 0.0123456790123457*M**2*S*T*epsilon**3 + 0.0185185185185185*M**2*S*T*epsilon**2 - 0.00925925925925926*M**2*S*T*epsilon + 0.00154320987654321*M**2*S*T - 0.00617283950617284*M**2*T**2*epsilon**3 + 0.00925925925925926*M**2*T**2*epsilon**2 - 0.00462962962962963*M**2*T**2*epsilon + 0.000771604938271605*M**2*T**2 + 0.0679012345679012*M*P**3*epsilon**3 - 0.058641975308642*M*P**3*epsilon**2 + 0.0123456790123457*M*P**3*epsilon + 0.104938271604938*M*P**2*R*epsilon**3 - 0.0648148148148148*M*P**2*R*epsilon**2 + 0.00308641975308642*M*P**2*R - 0.117283950617284*M*P**2*S*epsilon**3 + 0.0709876543209877*M*P**2*S*epsilon**2 + 0.00308641975308642*M*P**2*S*epsilon - 0.00462962962962963*M*P**2*S - 0.179012345679012*M*P**2*T*epsilon**3 + 0.151234567901235*M*P**2*T*epsilon**2 - 0.0308641975308642*M*P**2*T*epsilon + 0.037037037037037*M*P*R**2*epsilon**3 - 0.0138888888888889*M*P*R**2*epsilon + 0.00231481481481481*M*P*R**2 - 0.0864197530864197*M*P*R*S*epsilon**3 - 0.00617283950617284*M*P*R*S*epsilon**2 + 0.0432098765432099*M*P*R*S*epsilon - 0.00925925925925926*M*P*R*S - 0.172839506172839*M*P*R*T*epsilon**3 + 0.0987654320987654*M*P*R*T*epsilon**2 + 0.00308641975308642*M*P*R*T*epsilon - 0.00462962962962963*M*P*R*T + 0.0493827160493827*M*P*S**2*epsilon**3 + 0.00617283950617284*M*P*S**2*epsilon**2 - 0.029320987654321*M*P*S**2*epsilon + 0.00694444444444444*M*P*S**2 + 0.197530864197531*M*P*S*T*epsilon**3 - 0.111111111111111*M*P*S*T*epsilon**2 - 0.00925925925925926*M*P*S*T*epsilon + 0.00771604938271605*M*P*S*T + 0.154320987654321*M*P*T**2*epsilon**3 - 0.126543209876543*M*P*T**2*epsilon**2 + 0.0246913580246914*M*P*T**2*epsilon + 0.00617283950617284*M*R**3*epsilon**2 - 0.00154320987654321*M*R**3*epsilon - 0.000771604938271605*M*R**3 - 0.0308641975308642*M*R**2*S*epsilon**2 + 0.0169753086419753*M*R**2*S*epsilon - 0.000771604938271605*M*R**2*S - 0.0246913580246914*M*R**2*T*epsilon**3 - 0.00617283950617284*M*R**2*T*epsilon**2 + 0.0108024691358025*M*R**2*T*epsilon - 0.000771604938271605*M*R**2*T + 0.0432098765432099*M*R*S**2*epsilon**2 - 0.029320987654321*M*R*S**2*epsilon + 0.00385802469135802*M*R*S**2 + 0.0617283950617284*M*R*S*T*epsilon**3 + 0.0185185185185185*M*R*S*T*epsilon**2 - 0.037037037037037*M*R*S*T*epsilon + 0.00617283950617284*M*R*S*T + 0.0679012345679012*M*R*T**2*epsilon**3 - 0.0339506172839506*M*R*T**2*epsilon**2 - 0.00308641975308642*M*R*T**2*epsilon + 0.00154320987654321*M*R*T**2 - 0.0185185185185185*M*S**3*epsilon**2 + 0.0138888888888889*M*S**3*epsilon - 0.00231481481481481*M*S**3 - 0.037037037037037*M*S**2*T*epsilon**3 - 0.0123456790123457*M*S**2*T*epsilon**2 + 0.0262345679012346*M*S**2*T*epsilon - 0.00540123456790123*M*S**2*T - 0.0802469135802469*M*S*T**2*epsilon**3 + 0.0401234567901235*M*S*T**2*epsilon**2 + 0.00617283950617284*M*S*T**2*epsilon - 0.00308641975308642*M*S*T**2 - 0.0432098765432099*M*T**3*epsilon**3 + 0.0339506172839506*M*T**3*epsilon**2 - 0.00617283950617284*M*T**3*epsilon - 0.0925925925925926*P**4*epsilon**3 + 0.0462962962962963*P**4*epsilon**2 - 0.12962962962963*P**3*R*epsilon**3 + 0.0277777777777778*P**3*R*epsilon**2 + 0.0185185185185185*P**3*R*epsilon + 0.154320987654321*P**3*S*epsilon**3 - 0.0154320987654321*P**3*S*epsilon**2 - 0.0308641975308642*P**3*S*epsilon + 0.277777777777778*P**3*T*epsilon**3 - 0.138888888888889*P**3*T*epsilon**2 - 0.037037037037037*P**2*R**2*epsilon**3 - 0.0246913580246914*P**2*R**2*epsilon**2 + 0.0200617283950617*P**2*R**2*epsilon + 0.000771604938271605*P**2*R**2 + 0.0987654320987654*P**2*R*S*epsilon**3 + 0.0864197530864197*P**2*R*S*epsilon**2 - 0.058641975308642*P**2*R*S*epsilon - 0.00462962962962963*P**2*R*S + 0.259259259259259*P**2*R*T*epsilon**3 - 0.0555555555555556*P**2*R*T*epsilon**2 - 0.037037037037037*P**2*R*T*epsilon - 0.0617283950617284*P**2*S**2*epsilon**3 - 0.0679012345679012*P**2*S**2*epsilon**2 + 0.0401234567901235*P**2*S**2*epsilon + 0.00462962962962963*P**2*S**2 - 0.320987654320988*P**2*S*T*epsilon**3 + 0.0246913580246914*P**2*S*T*epsilon**2 + 0.0679012345679012*P**2*S*T*epsilon - 0.296296296296296*P**2*T**2*epsilon**3 + 0.148148148148148*P**2*T**2*epsilon**2 - 0.00617283950617284*P*R**3*epsilon**2 + 0.00154320987654321*P*R**3*epsilon + 0.000771604938271605*P*R**3 + 0.0432098765432099*P*R**2*S*epsilon**2 - 0.0108024691358025*P*R**2*S*epsilon - 0.00540123456790123*P*R**2*S + 0.037037037037037*P*R**2*T*epsilon**3 + 0.0246913580246914*P*R**2*T*epsilon**2 - 0.0200617283950617*P*R**2*T*epsilon - 0.000771604938271605*P*R**2*T - 0.0740740740740741*P*R*S**2*epsilon**2 + 0.0138888888888889*P*R*S**2*epsilon + 0.0115740740740741*P*R*S**2 - 0.111111111111111*P*R*S*T*epsilon**3 - 0.104938271604938*P*R*S*T*epsilon**2 + 0.0679012345679012*P*R*S*T*epsilon + 0.00617283950617284*P*R*S*T - 0.154320987654321*P*R*T**2*epsilon**3 + 0.0339506172839506*P*R*T**2*epsilon**2 + 0.0216049382716049*P*R*T**2*epsilon + 0.037037037037037*P*S**3*epsilon**2 - 0.00462962962962963*P*S**3*epsilon - 0.00694444444444444*P*S**3 + 0.0740740740740741*P*S**2*T*epsilon**3 + 0.0925925925925926*P*S**2*T*epsilon**2 - 0.0509259259259259*P*S**2*T*epsilon - 0.00694444444444444*P*S**2*T + 0.203703703703704*P*S*T**2*epsilon**3 - 0.00925925925925926*P*S*T**2*epsilon**2 - 0.0462962962962963*P*S*T**2*epsilon + 0.12962962962963*P*T**3*epsilon**3 - 0.0648148148148148*P*T**3*epsilon**2 - 0.00462962962962963*R**2*S**2*epsilon + 0.00231481481481481*R**2*S**2 - 0.0123456790123457*R**2*S*T*epsilon**2 + 0.00308641975308642*R**2*S*T*epsilon + 0.00154320987654321*R**2*S*T - 0.00617283950617284*R**2*T**2*epsilon**3 - 0.00308641975308642*R**2*T**2*epsilon**2 + 0.00308641975308642*R**2*T**2*epsilon + 0.00925925925925926*R*S**3*epsilon - 0.00462962962962963*R*S**3 + 0.0308641975308642*R*S**2*T*epsilon**2 - 0.00308641975308642*R*S**2*T*epsilon - 0.00617283950617284*R*S**2*T + 0.0246913580246914*R*S*T**2*epsilon**3 + 0.0246913580246914*R*S*T**2*epsilon**2 - 0.0154320987654321*R*S*T**2*epsilon - 0.00154320987654321*R*S*T**2 + 0.0246913580246914*R*T**3*epsilon**3 - 0.00617283950617284*R*T**3*epsilon**2 - 0.00308641975308642*R*T**3*epsilon - 0.00462962962962963*S**4*epsilon + 0.00231481481481481*S**4 - 0.0185185185185185*S**3*T*epsilon**2 + 0.00462962962962963*S**3*T - 0.0185185185185185*S**2*T**2*epsilon**3 - 0.0277777777777778*S**2*T**2*epsilon**2 + 0.0138888888888889*S**2*T**2*epsilon + 0.00231481481481481*S**2*T**2 - 0.037037037037037*S*T**3*epsilon**3 + 0.00925925925925926*S*T**3*epsilon - 0.0185185185185185*T**4*epsilon**3 + 0.00925925925925926*T**4*epsilon**2) + UGA_star**4*UgA_star*(0.0123456790123457*M**2*P**2*epsilon**3 - 0.0185185185185185*M**2*P**2*epsilon**2 + 0.00925925925925926*M**2*P**2*epsilon - 0.00154320987654321*M**2*P**2 + 0.0123456790123457*M**2*P*R*epsilon**3 - 0.0185185185185185*M**2*P*R*epsilon**2 + 0.00925925925925926*M**2*P*R*epsilon - 0.00154320987654321*M**2*P*R - 0.0185185185185185*M**2*P*S*epsilon**3 + 0.0277777777777778*M**2*P*S*epsilon**2 - 0.0138888888888889*M**2*P*S*epsilon + 0.00231481481481481*M**2*P*S - 0.0185185185185185*M**2*P*T*epsilon**3 + 0.0277777777777778*M**2*P*T*epsilon**2 - 0.0138888888888889*M**2*P*T*epsilon + 0.00231481481481481*M**2*P*T - 0.00617283950617284*M**2*R*S*epsilon**3 + 0.00925925925925926*M**2*R*S*epsilon**2 - 0.00462962962962963*M**2*R*S*epsilon + 0.000771604938271605*M**2*R*S - 0.00617283950617284*M**2*R*T*epsilon**3 + 0.00925925925925926*M**2*R*T*epsilon**2 - 0.00462962962962963*M**2*R*T*epsilon + 0.000771604938271605*M**2*R*T + 0.00617283950617284*M**2*S**2*epsilon**3 - 0.00925925925925926*M**2*S**2*epsilon**2 + 0.00462962962962963*M**2*S**2*epsilon - 0.000771604938271605*M**2*S**2 + 0.0123456790123457*M**2*S*T*epsilon**3 - 0.0185185185185185*M**2*S*T*epsilon**2 + 0.00925925925925926*M**2*S*T*epsilon - 0.00154320987654321*M**2*S*T + 0.00617283950617284*M**2*T**2*epsilon**3 - 0.00925925925925926*M**2*T**2*epsilon**2 + 0.00462962962962963*M**2*T**2*epsilon - 0.000771604938271605*M**2*T**2 - 0.135802469135802*M*P**3*epsilon**3 + 0.117283950617284*M*P**3*epsilon**2 - 0.0246913580246914*M*P**3*epsilon - 0.0740740740740741*M*P**2*R*epsilon**3 + 0.0123456790123457*M*P**2*R*epsilon**2 + 0.0246913580246914*M*P**2*R*epsilon - 0.00617283950617284*M*P**2*R + 0.166666666666667*M*P**2*S*epsilon**3 - 0.0833333333333333*M*P**2*S*epsilon**2 - 0.0185185185185185*M*P**2*S*epsilon + 0.00925925925925926*M*P**2*S + 0.290123456790123*M*P**2*T*epsilon**3 - 0.243827160493827*M*P**2*T*epsilon**2 + 0.0493827160493827*M*P**2*T*epsilon - 0.0123456790123457*M*P*R**2*epsilon**2 + 0.00308641975308642*M*P*R**2*epsilon + 0.00154320987654321*M*P*R**2 + 0.037037037037037*M*P*R*S*epsilon**3 + 0.0432098765432099*M*P*R*S*epsilon**2 - 0.0432098765432099*M*P*R*S*epsilon + 0.00617283950617284*M*P*R*S + 0.0864197530864197*M*P*R*T*epsilon**3 - 0.00617283950617284*M*P*R*T*epsilon**2 - 0.0308641975308642*M*P*R*T*epsilon + 0.00617283950617284*M*P*R*T - 0.0493827160493827*M*P*S**2*epsilon**3 - 0.0246913580246914*M*P*S**2*epsilon**2 + 0.0432098765432099*M*P*S**2*epsilon - 0.00925925925925926*M*P*S**2 - 0.234567901234568*M*P*S*T*epsilon**3 + 0.117283950617284*M*P*S*T*epsilon**2 + 0.0216049382716049*M*P*S*T*epsilon - 0.0108024691358025*M*P*S*T - 0.197530864197531*M*P*T**2*epsilon**3 + 0.160493827160494*M*P*T**2*epsilon**2 - 0.0308641975308642*M*P*T**2*epsilon + 0.00617283950617284*M*R**2*S*epsilon**2 - 0.00154320987654321*M*R**2*S*epsilon - 0.000771604938271605*M*R**2*S + 0.00617283950617284*M*R**2*T*epsilon**2 - 0.00154320987654321*M*R**2*T*epsilon - 0.000771604938271605*M*R**2*T - 0.0246913580246914*M*R*S**2*epsilon**2 + 0.0154320987654321*M*R*S**2*epsilon - 0.00154320987654321*M*R*S**2 - 0.0246913580246914*M*R*S*T*epsilon**3 - 0.0246913580246914*M*R*S*T*epsilon**2 + 0.0246913580246914*M*R*S*T*epsilon - 0.00308641975308642*M*R*S*T - 0.0246913580246914*M*R*T**2*epsilon**3 + 0.00925925925925926*M*R*T**2*epsilon - 0.00154320987654321*M*R*T**2 + 0.0185185185185185*M*S**3*epsilon**2 - 0.0138888888888889*M*S**3*epsilon + 0.00231481481481481*M*S**3 + 0.037037037037037*M*S**2*T*epsilon**3 + 0.0123456790123457*M*S**2*T*epsilon**2 - 0.0262345679012346*M*S**2*T*epsilon + 0.00540123456790123*M*S**2*T + 0.0802469135802469*M*S*T**2*epsilon**3 - 0.0401234567901235*M*S*T**2*epsilon**2 - 0.00617283950617284*M*S*T**2*epsilon + 0.00308641975308642*M*S*T**2 + 0.0432098765432099*M*T**3*epsilon**3 - 0.0339506172839506*M*T**3*epsilon**2 + 0.00617283950617284*M*T**3*epsilon + 0.185185185185185*P**4*epsilon**3 - 0.0925925925925926*P**4*epsilon**2 + 0.0740740740740741*P**3*R*epsilon**3 + 0.037037037037037*P**3*R*epsilon**2 - 0.037037037037037*P**3*R*epsilon - 0.216049382716049*P**3*S*epsilon**3 - 0.0154320987654321*P**3*S*epsilon**2 + 0.0617283950617284*P**3*S*epsilon - 0.462962962962963*P**3*T*epsilon**3 + 0.231481481481481*P**3*T*epsilon**2 + 0.0123456790123457*P**2*R**2*epsilon**2 - 0.00308641975308642*P**2*R**2*epsilon - 0.00154320987654321*P**2*R**2 - 0.037037037037037*P**2*R*S*epsilon**3 - 0.0925925925925926*P**2*R*S*epsilon**2 + 0.037037037037037*P**2*R*S*epsilon + 0.00925925925925926*P**2*R*S - 0.111111111111111*P**2*R*T*epsilon**3 - 0.0555555555555556*P**2*R*T*epsilon**2 + 0.0555555555555556*P**2*R*T*epsilon + 0.0617283950617284*P**2*S**2*epsilon**3 + 0.104938271604938*P**2*S**2*epsilon**2 - 0.0493827160493827*P**2*S**2*epsilon - 0.00925925925925926*P**2*S**2 + 0.395061728395062*P**2*S*T*epsilon**3 + 0.0123456790123457*P**2*S*T*epsilon**2 - 0.104938271604938*P**2*S*T*epsilon + 0.407407407407407*P**2*T**2*epsilon**3 - 0.203703703703704*P**2*T**2*epsilon**2 - 0.00617283950617284*P*R**2*S*epsilon**2 + 0.00154320987654321*P*R**2*S*epsilon + 0.000771604938271605*P*R**2*S - 0.00617283950617284*P*R**2*T*epsilon**2 + 0.00154320987654321*P*R**2*T*epsilon + 0.000771604938271605*P*R**2*T + 0.037037037037037*P*R*S**2*epsilon**2 - 0.00925925925925926*P*R*S**2 + 0.037037037037037*P*R*S*T*epsilon**3 + 0.0802469135802469*P*R*S*T*epsilon**2 - 0.0339506172839506*P*R*S*T*epsilon - 0.00771604938271605*P*R*S*T + 0.0493827160493827*P*R*T**2*epsilon**3 + 0.0246913580246914*P*R*T**2*epsilon**2 - 0.0246913580246914*P*R*T**2*epsilon - 0.037037037037037*P*S**3*epsilon**2 + 0.00925925925925926*P*S**3 - 0.0740740740740741*P*S**2*T*epsilon**3 - 0.111111111111111*P*S**2*T*epsilon**2 + 0.0555555555555556*P*S**2*T*epsilon + 0.00925925925925926*P*S**2*T - 0.222222222222222*P*S*T**2*epsilon**3 + 0.0555555555555556*P*S*T**2*epsilon - 0.148148148148148*P*T**3*epsilon**3 + 0.0740740740740741*P*T**3*epsilon**2 - 0.00462962962962963*R*S**3*epsilon + 0.00231481481481481*R*S**3 - 0.0123456790123457*R*S**2*T*epsilon**2 - 0.00154320987654321*R*S**2*T*epsilon + 0.00385802469135802*R*S**2*T - 0.00617283950617284*R*S*T**2*epsilon**3 - 0.0154320987654321*R*S*T**2*epsilon**2 + 0.00617283950617284*R*S*T**2*epsilon + 0.00154320987654321*R*S*T**2 - 0.00617283950617284*R*T**3*epsilon**3 - 0.00308641975308642*R*T**3*epsilon**2 + 0.00308641975308642*R*T**3*epsilon + 0.00462962962962963*S**4*epsilon - 0.00231481481481481*S**4 + 0.0185185185185185*S**3*T*epsilon**2 - 0.00462962962962963*S**3*T + 0.0185185185185185*S**2*T**2*epsilon**3 + 0.0277777777777778*S**2*T**2*epsilon**2 - 0.0138888888888889*S**2*T**2*epsilon - 0.00231481481481481*S**2*T**2 + 0.037037037037037*S*T**3*epsilon**3 - 0.00925925925925926*S*T**3*epsilon + 0.0185185185185185*T**4*epsilon**3 - 0.00925925925925926*T**4*epsilon**2) + UGA_star**4*(-0.0123456790123457*M**3*P*epsilon**3 + 0.0185185185185185*M**3*P*epsilon**2 - 0.00925925925925926*M**3*P*epsilon + 0.00154320987654321*M**3*P - 0.0123456790123457*M**3*R*epsilon**3 + 0.0185185185185185*M**3*R*epsilon**2 - 0.00925925925925926*M**3*R*epsilon + 0.00154320987654321*M**3*R + 0.0123456790123457*M**3*S*epsilon**3 - 0.0185185185185185*M**3*S*epsilon**2 + 0.00925925925925926*M**3*S*epsilon - 0.00154320987654321*M**3*S + 0.0123456790123457*M**3*T*epsilon**3 - 0.0185185185185185*M**3*T*epsilon**2 + 0.00925925925925926*M**3*T*epsilon - 0.00154320987654321*M**3*T + 0.216049382716049*M**2*P**2*epsilon**3 - 0.212962962962963*M**2*P**2*epsilon**2 + 0.0601851851851852*M**2*P**2*epsilon - 0.00385802469135802*M**2*P**2 + 0.141975308641975*M**2*P*R*epsilon**3 - 0.0833333333333333*M**2*P*R*epsilon**2 - 0.00462962962962963*M**2*P*R*epsilon + 0.00540123456790123*M**2*P*R - 0.216049382716049*M**2*P*S*epsilon**3 + 0.157407407407407*M**2*P*S*epsilon**2 - 0.0138888888888889*M**2*P*S*epsilon - 0.00540123456790123*M**2*P*S - 0.327160493827161*M**2*P*T*epsilon**3 + 0.305555555555556*M**2*P*T*epsilon**2 - 0.0787037037037037*M**2*P*T*epsilon + 0.00385802469135802*M**2*P*T + 0.00617283950617284*M**2*R**2*epsilon**3 + 0.00925925925925926*M**2*R**2*epsilon**2 - 0.00462962962962963*M**2*R**2*epsilon - 0.000771604938271605*M**2*R**2 - 0.037037037037037*M**2*R*S*epsilon**3 - 0.0185185185185185*M**2*R*S*epsilon**2 + 0.0277777777777778*M**2*R*S*epsilon - 0.00462962962962963*M**2*R*S - 0.0864197530864197*M**2*R*T*epsilon**3 + 0.037037037037037*M**2*R*T*epsilon**2 + 0.00925925925925926*M**2*R*T*epsilon - 0.00308641975308642*M**2*R*T + 0.037037037037037*M**2*S**2*epsilon**3 - 0.0185185185185185*M**2*S**2*epsilon + 0.00462962962962963*M**2*S**2 + 0.148148148148148*M**2*S*T*epsilon**3 - 0.0925925925925926*M**2*S*T*epsilon**2 + 0.00462962962962963*M**2*S*T + 0.117283950617284*M**2*T**2*epsilon**3 - 0.101851851851852*M**2*T**2*epsilon**2 + 0.0231481481481481*M**2*T**2*epsilon - 0.000771604938271605*M**2*T**2 - 0.549382716049383*M*P**3*epsilon**3 + 0.373456790123457*M*P**3*epsilon**2 - 0.0493827160493827*M*P**3*epsilon - 0.283950617283951*M*P**2*R*epsilon**3 + 0.0679012345679012*M*P**2*R*epsilon**2 + 0.0432098765432099*M*P**2*R*epsilon - 0.00308641975308642*M*P**2*R + 0.567901234567901*M*P**2*S*epsilon**3 - 0.172839506172839*M*P**2*S*epsilon**2 - 0.0771604938271605*M*P**2*S*epsilon + 0.0108024691358025*M*P**2*S + 1.0*M*P**2*T*epsilon**3 - 0.648148148148148*M*P**2*T*epsilon**2 + 0.0740740740740741*M*P**2*T*epsilon - 0.0123456790123457*M*P*R**2*epsilon**3 - 0.0123456790123457*M*P*R**2*epsilon**2 + 0.00771604938271605*M*P*R**2*epsilon + 0.000771604938271605*M*P*R**2 + 0.0987654320987654*M*P*R*S*epsilon**3 + 0.0864197530864197*M*P*R*S*epsilon**2 - 0.058641975308642*M*P*R*S*epsilon - 0.00462962962962963*M*P*R*S + 0.246913580246914*M*P*R*T*epsilon**3 - 0.037037037037037*M*P*R*T*epsilon**2 - 0.0462962962962963*M*P*R*T*epsilon + 0.00154320987654321*M*P*R*T - 0.123456790123457*M*P*S**2*epsilon**3 - 0.0925925925925926*M*P*S**2*epsilon**2 + 0.0787037037037037*M*P*S**2*epsilon - 0.000771604938271605*M*P*S**2 - 0.604938271604938*M*P*S*T*epsilon**3 + 0.141975308641975*M*P*S*T*epsilon**2 + 0.0987654320987654*M*P*S*T*epsilon - 0.00925925925925926*M*P*S*T - 0.549382716049383*M*P*T**2*epsilon**3 + 0.33641975308642*M*P*T**2*epsilon**2 - 0.0308641975308642*M*P*T**2*epsilon - 0.0185185185185185*M*R*S**2*epsilon**2 + 0.00462962962962963*M*R*S**2 - 0.0246913580246914*M*R*S*T*epsilon**3 - 0.037037037037037*M*R*S*T*epsilon**2 + 0.0185185185185185*M*R*S*T*epsilon + 0.00308641975308642*M*R*S*T - 0.037037037037037*M*R*T**2*epsilon**3 + 0.00925925925925926*M*R*T**2*epsilon + 0.0246913580246914*M*S**3*epsilon**2 - 0.00617283950617284*M*S**3*epsilon - 0.00308641975308642*M*S**3 + 0.0493827160493827*M*S**2*T*epsilon**3 + 0.0555555555555556*M*S**2*T*epsilon**2 - 0.037037037037037*M*S**2*T*epsilon - 0.00154320987654321*M*S**2*T + 0.135802469135802*M*S*T**2*epsilon**3 - 0.0185185185185185*M*S*T**2*epsilon**2 - 0.0277777777777778*M*S*T**2*epsilon + 0.00154320987654321*M*S*T**2 + 0.0864197530864197*M*T**3*epsilon**3 - 0.0493827160493827*M*T**3*epsilon**2 + 0.00308641975308642*M*T**3*epsilon + 0.339506172839506*P**4*epsilon**3 - 0.169753086419753*P**4*epsilon**2 + 0.148148148148148*P**3*R*epsilon**3 - 0.0185185185185185*P**3*R*epsilon**2 - 0.0277777777777778*P**3*R*epsilon - 0.37037037037037*P**3*S*epsilon**3 + 0.0308641975308642*P**3*S*epsilon**2 + 0.0771604938271605*P**3*S*epsilon - 0.679012345679012*P**3*T*epsilon**3 + 0.339506172839506*P**3*T*epsilon**2 + 0.00617283950617284*P**2*R**2*epsilon**3 + 0.00308641975308642*P**2*R**2*epsilon**2 - 0.00308641975308642*P**2*R**2*epsilon - 0.0617283950617284*P**2*R*S*epsilon**3 - 0.0555555555555556*P**2*R*S*epsilon**2 + 0.037037037037037*P**2*R*S*epsilon + 0.00308641975308642*P**2*R*S - 0.148148148148148*P**2*R*T*epsilon**3 + 0.0185185185185185*P**2*R*T*epsilon**2 + 0.0277777777777778*P**2*R*T*epsilon + 0.0925925925925926*P**2*S**2*epsilon**3 + 0.101851851851852*P**2*S**2*epsilon**2 - 0.0601851851851852*P**2*S**2*epsilon - 0.00694444444444444*P**2*S**2 + 0.481481481481481*P**2*S*T*epsilon**3 - 0.037037037037037*P**2*S*T*epsilon**2 - 0.101851851851852*P**2*S*T*epsilon + 0.444444444444444*P**2*T**2*epsilon**3 - 0.222222222222222*P**2*T**2*epsilon**2 + 0.0185185185185185*P*R*S**2*epsilon**2 - 0.00462962962962963*P*R*S**2*epsilon - 0.00231481481481481*P*R*S**2 + 0.0246913580246914*P*R*S*T*epsilon**3 + 0.0246913580246914*P*R*S*T*epsilon**2 - 0.0154320987654321*P*R*S*T*epsilon - 0.00154320987654321*P*R*S*T + 0.0308641975308642*P*R*T**2*epsilon**3 - 0.00308641975308642*P*R*T**2*epsilon**2 - 0.00617283950617284*P*R*T**2*epsilon - 0.0308641975308642*P*S**3*epsilon**2 + 0.00462962962962963*P*S**3*epsilon + 0.00540123456790123*P*S**3 - 0.0617283950617284*P*S**2*T*epsilon**3 - 0.0740740740740741*P*S**2*T*epsilon**2 + 0.0416666666666667*P*S**2*T*epsilon + 0.00540123456790123*P*S**2*T - 0.166666666666667*P*S*T**2*epsilon**3 + 0.00925925925925926*P*S*T**2*epsilon**2 + 0.037037037037037*P*S*T**2*epsilon - 0.104938271604938*P*T**3*epsilon**3 + 0.0524691358024691*P*T**3*epsilon**2 + 0.00154320987654321*S**4*epsilon - 0.000771604938271605*S**4 + 0.00617283950617284*S**3*T*epsilon**2 - 0.00154320987654321*S**3*T + 0.00617283950617284*S**2*T**2*epsilon**3 + 0.00925925925925926*S**2*T**2*epsilon**2 - 0.00462962962962963*S**2*T**2*epsilon - 0.000771604938271605*S**2*T**2 + 0.0123456790123457*S*T**3*epsilon**3 - 0.00308641975308642*S*T**3*epsilon + 0.00617283950617284*T**4*epsilon**3 - 0.00308641975308642*T**4*epsilon**2) + UGA_star**3*UgA_star**2*(0.037037037037037*M**2*P**2*epsilon**3 - 0.0432098765432099*M**2*P**2*epsilon**2 + 0.0154320987654321*M**2*P**2*epsilon - 0.00154320987654321*M**2*P**2 + 0.0493827160493827*M**2*P*R*epsilon**3 - 0.0493827160493827*M**2*P*R*epsilon**2 + 0.0123456790123457*M**2*P*R*epsilon - 0.0555555555555556*M**2*P*S*epsilon**3 + 0.058641975308642*M**2*P*S*epsilon**2 - 0.0169753086419753*M**2*P*S*epsilon + 0.000771604938271605*M**2*P*S - 0.0679012345679012*M**2*P*T*epsilon**3 + 0.0771604938271605*M**2*P*T*epsilon**2 - 0.0262345679012346*M**2*P*T*epsilon + 0.00231481481481481*M**2*P*T + 0.0123456790123457*M**2*R**2*epsilon**3 - 0.00617283950617284*M**2*R**2*epsilon**2 - 0.00308641975308642*M**2*R**2*epsilon + 0.00154320987654321*M**2*R**2 - 0.0308641975308642*M**2*R*S*epsilon**3 + 0.0216049382716049*M**2*R*S*epsilon**2 + 0.00154320987654321*M**2*R*S*epsilon - 0.00231481481481481*M**2*R*S - 0.0432098765432099*M**2*R*T*epsilon**3 + 0.0401234567901235*M**2*R*T*epsilon**2 - 0.00771604938271605*M**2*R*T*epsilon - 0.000771604938271605*M**2*R*T + 0.0185185185185185*M**2*S**2*epsilon**3 - 0.0154320987654321*M**2*S**2*epsilon**2 + 0.00154320987654321*M**2*S**2*epsilon + 0.000771604938271605*M**2*S**2 + 0.0493827160493827*M**2*S*T*epsilon**3 - 0.0493827160493827*M**2*S*T*epsilon**2 + 0.0123456790123457*M**2*S*T*epsilon + 0.0308641975308642*M**2*T**2*epsilon**3 - 0.0339506172839506*M**2*T**2*epsilon**2 + 0.0108024691358025*M**2*T**2*epsilon - 0.000771604938271605*M**2*T**2 - 0.148148148148148*M*P**3*epsilon**3 + 0.111111111111111*M*P**3*epsilon**2 - 0.0185185185185185*M*P**3*epsilon - 0.185185185185185*M*P**2*R*epsilon**3 + 0.104938271604938*M*P**2*R*epsilon**2 - 0.00308641975308642*M*P**2*R*epsilon - 0.00154320987654321*M*P**2*R + 0.222222222222222*M*P**2*S*epsilon**3 - 0.111111111111111*M*P**2*S*epsilon**2 - 0.00925925925925926*M*P**2*S*epsilon + 0.00462962962962963*M*P**2*S + 0.333333333333333*M*P**2*T*epsilon**3 - 0.240740740740741*M*P**2*T*epsilon**2 + 0.037037037037037*M*P**2*T*epsilon - 0.037037037037037*M*P*R**2*epsilon**3 - 0.00617283950617284*M*P*R**2*epsilon**2 + 0.0154320987654321*M*P*R**2*epsilon - 0.00154320987654321*M*P*R**2 + 0.111111111111111*M*P*R*S*epsilon**3 + 0.0185185185185185*M*P*R*S*epsilon**2 - 0.0462962962962963*M*P*R*S*epsilon + 0.00462962962962963*M*P*R*S + 0.234567901234568*M*P*R*T*epsilon**3 - 0.117283950617284*M*P*R*T*epsilon**2 - 0.00308641975308642*M*P*R*T*epsilon + 0.00154320987654321*M*P*R*T - 0.0740740740740741*M*P*S**2*epsilon**3 - 0.0185185185185185*M*P*S**2*epsilon**2 + 0.037037037037037*M*P*S**2*epsilon - 0.00462962962962963*M*P*S**2 - 0.296296296296296*M*P*S*T*epsilon**3 + 0.123456790123457*M*P*S*T*epsilon**2 + 0.0246913580246914*M*P*S*T*epsilon - 0.00617283950617284*M*P*S*T - 0.234567901234568*M*P*T**2*epsilon**3 + 0.160493827160494*M*P*T**2*epsilon**2 - 0.0216049382716049*M*P*T**2*epsilon + 0.0123456790123457*M*R**2*S*epsilon**2 - 0.00308641975308642*M*R**2*S*epsilon - 0.00154320987654321*M*R**2*S + 0.0123456790123457*M*R**2*T*epsilon**3 + 0.00617283950617284*M*R**2*T*epsilon**2 - 0.00617283950617284*M*R**2*T*epsilon - 0.0308641975308642*M*R*S**2*epsilon**2 + 0.0123456790123457*M*R*S**2*epsilon + 0.00154320987654321*M*R*S**2 - 0.0493827160493827*M*R*S*T*epsilon**3 - 0.0246913580246914*M*R*S*T*epsilon**2 + 0.0246913580246914*M*R*S*T*epsilon - 0.0617283950617284*M*R*T**2*epsilon**3 + 0.0246913580246914*M*R*T**2*epsilon**2 + 0.00308641975308642*M*R*T**2*epsilon + 0.0185185185185185*M*S**3*epsilon**2 - 0.00925925925925926*M*S**3*epsilon + 0.037037037037037*M*S**2*T*epsilon**3 + 0.0246913580246914*M*S**2*T*epsilon**2 - 0.0246913580246914*M*S**2*T*epsilon + 0.00154320987654321*M*S**2*T + 0.0864197530864197*M*S*T**2*epsilon**3 - 0.0246913580246914*M*S*T**2*epsilon**2 - 0.0123456790123457*M*S*T**2*epsilon + 0.00154320987654321*M*S*T**2 + 0.0493827160493827*M*T**3*epsilon**3 - 0.0308641975308642*M*T**3*epsilon**2 + 0.00308641975308642*M*T**3*epsilon + 0.123456790123457*P**4*epsilon**3 - 0.0617283950617284*P**4*epsilon**2 + 0.148148148148148*P**3*R*epsilon**3 - 0.0493827160493827*P**3*R*epsilon**2 - 0.0123456790123457*P**3*R*epsilon - 0.185185185185185*P**3*S*epsilon**3 + 0.0308641975308642*P**3*S*epsilon**2 + 0.0308641975308642*P**3*S*epsilon - 0.308641975308642*P**3*T*epsilon**3 + 0.154320987654321*P**3*T*epsilon**2 + 0.0246913580246914*P**2*R**2*epsilon**3 + 0.0123456790123457*P**2*R**2*epsilon**2 - 0.0123456790123457*P**2*R**2*epsilon - 0.0864197530864197*P**2*R*S*epsilon**3 - 0.0555555555555556*P**2*R*S*epsilon**2 + 0.0462962962962963*P**2*R*S*epsilon + 0.00154320987654321*P**2*R*S - 0.222222222222222*P**2*R*T*epsilon**3 + 0.0740740740740741*P**2*R*T*epsilon**2 + 0.0185185185185185*P**2*R*T*epsilon + 0.0617283950617284*P**2*S**2*epsilon**3 + 0.0555555555555556*P**2*S**2*epsilon**2 - 0.037037037037037*P**2*S**2*epsilon - 0.00308641975308642*P**2*S**2 + 0.296296296296296*P**2*S*T*epsilon**3 - 0.037037037037037*P**2*S*T*epsilon**2 - 0.0555555555555556*P**2*S*T*epsilon + 0.259259259259259*P**2*T**2*epsilon**3 - 0.12962962962963*P**2*T**2*epsilon**2 - 0.0123456790123457*P*R**2*S*epsilon**2 + 0.00308641975308642*P*R**2*S*epsilon + 0.00154320987654321*P*R**2*S - 0.0123456790123457*P*R**2*T*epsilon**3 - 0.00617283950617284*P*R**2*T*epsilon**2 + 0.00617283950617284*P*R**2*T*epsilon + 0.037037037037037*P*R*S**2*epsilon**2 - 0.00925925925925926*P*R*S**2*epsilon - 0.00462962962962963*P*R*S**2 + 0.0617283950617284*P*R*S*T*epsilon**3 + 0.0432098765432099*P*R*S*T*epsilon**2 - 0.0339506172839506*P*R*S*T*epsilon - 0.00154320987654321*P*R*S*T + 0.0864197530864197*P*R*T**2*epsilon**3 - 0.0308641975308642*P*R*T**2*epsilon**2 - 0.00617283950617284*P*R*T**2*epsilon - 0.0246913580246914*P*S**3*epsilon**2 + 0.00462962962962963*P*S**3*epsilon + 0.00385802469135802*P*S**3 - 0.0493827160493827*P*S**2*T*epsilon**3 - 0.0555555555555556*P*S**2*T*epsilon**2 + 0.0324074074074074*P*S**2*T*epsilon + 0.00385802469135802*P*S**2*T - 0.12962962962963*P*S*T**2*epsilon**3 + 0.00925925925925926*P*S*T**2*epsilon**2 + 0.0277777777777778*P*S*T**2*epsilon - 0.0802469135802469*P*T**3*epsilon**3 + 0.0401234567901235*P*T**3*epsilon**2 - 0.00154320987654321*R*S**3*epsilon + 0.000771604938271605*R*S**3 - 0.00617283950617284*R*S**2*T*epsilon**2 + 0.00154320987654321*R*S**2*T*epsilon + 0.000771604938271605*R*S**2*T - 0.00617283950617284*R*S*T**2*epsilon**3 - 0.00308641975308642*R*S*T**2*epsilon**2 + 0.00308641975308642*R*S*T**2*epsilon - 0.00617283950617284*R*T**3*epsilon**3 + 0.00308641975308642*R*T**3*epsilon**2 + 0.00154320987654321*S**4*epsilon - 0.000771604938271605*S**4 + 0.00617283950617284*S**3*T*epsilon**2 - 0.00154320987654321*S**3*T + 0.00617283950617284*S**2*T**2*epsilon**3 + 0.00925925925925926*S**2*T**2*epsilon**2 - 0.00462962962962963*S**2*T**2*epsilon - 0.000771604938271605*S**2*T**2 + 0.0123456790123457*S*T**3*epsilon**3 - 0.00308641975308642*S*T**3*epsilon + 0.00617283950617284*T**4*epsilon**3 - 0.00308641975308642*T**4*epsilon**2) + UGA_star**3*UgA_star*(-0.0740740740740741*M**2*P**2*epsilon**3 + 0.0864197530864197*M**2*P**2*epsilon**2 - 0.0308641975308642*M**2*P**2*epsilon + 0.00308641975308642*M**2*P**2 - 0.0246913580246914*M**2*P*R*epsilon**3 + 0.0123456790123457*M**2*P*R*epsilon**2 + 0.00617283950617284*M**2*P*R*epsilon - 0.00308641975308642*M**2*P*R + 0.0740740740740741*M**2*P*S*epsilon**3 - 0.0740740740740741*M**2*P*S*epsilon**2 + 0.0185185185185185*M**2*P*S*epsilon + 0.0987654320987654*M**2*P*T*epsilon**3 - 0.111111111111111*M**2*P*T*epsilon**2 + 0.037037037037037*M**2*P*T*epsilon - 0.00308641975308642*M**2*P*T + 0.0123456790123457*M**2*R*S*epsilon**3 - 0.00617283950617284*M**2*R*S*epsilon**2 - 0.00308641975308642*M**2*R*S*epsilon + 0.00154320987654321*M**2*R*S + 0.0123456790123457*M**2*R*T*epsilon**3 - 0.00617283950617284*M**2*R*T*epsilon**2 - 0.00308641975308642*M**2*R*T*epsilon + 0.00154320987654321*M**2*R*T - 0.0185185185185185*M**2*S**2*epsilon**3 + 0.0154320987654321*M**2*S**2*epsilon**2 - 0.00154320987654321*M**2*S**2*epsilon - 0.000771604938271605*M**2*S**2 - 0.0493827160493827*M**2*S*T*epsilon**3 + 0.0493827160493827*M**2*S*T*epsilon**2 - 0.0123456790123457*M**2*S*T*epsilon - 0.0308641975308642*M**2*T**2*epsilon**3 + 0.0339506172839506*M**2*T**2*epsilon**2 - 0.0108024691358025*M**2*T**2*epsilon + 0.000771604938271605*M**2*T**2 + 0.296296296296296*M*P**3*epsilon**3 - 0.222222222222222*M*P**3*epsilon**2 + 0.037037037037037*M*P**3*epsilon + 0.0740740740740741*M*P**2*R*epsilon**3 + 0.0123456790123457*M*P**2*R*epsilon**2 - 0.0308641975308642*M*P**2*R*epsilon + 0.00308641975308642*M*P**2*R - 0.296296296296296*M*P**2*S*epsilon**3 + 0.111111111111111*M*P**2*S*epsilon**2 + 0.037037037037037*M*P**2*S*epsilon - 0.00925925925925926*M*P**2*S - 0.518518518518518*M*P**2*T*epsilon**3 + 0.37037037037037*M*P**2*T*epsilon**2 - 0.0555555555555556*M*P**2*T*epsilon - 0.037037037037037*M*P*R*S*epsilon**3 - 0.0308641975308642*M*P*R*S*epsilon**2 + 0.0216049382716049*M*P*R*S*epsilon + 0.00154320987654321*M*P*R*S - 0.0617283950617284*M*P*R*T*epsilon**3 - 0.0185185185185185*M*P*R*T*epsilon**2 + 0.0277777777777778*M*P*R*T*epsilon - 0.00154320987654321*M*P*R*T + 0.0740740740740741*M*P*S**2*epsilon**3 + 0.037037037037037*M*P*S**2*epsilon**2 - 0.0462962962962963*M*P*S**2*epsilon + 0.00462962962962963*M*P*S**2 + 0.333333333333333*M*P*S*T*epsilon**3 - 0.117283950617284*M*P*S*T*epsilon**2 - 0.0401234567901235*M*P*S*T*epsilon + 0.00771604938271605*M*P*S*T + 0.283950617283951*M*P*T**2*epsilon**3 - 0.191358024691358*M*P*T**2*epsilon**2 + 0.0246913580246914*M*P*T**2*epsilon + 0.0123456790123457*M*R*S**2*epsilon**2 - 0.00308641975308642*M*R*S**2*epsilon - 0.00154320987654321*M*R*S**2 + 0.0123456790123457*M*R*S*T*epsilon**3 + 0.0185185185185185*M*R*S*T*epsilon**2 - 0.00925925925925926*M*R*S*T*epsilon - 0.00154320987654321*M*R*S*T + 0.0123456790123457*M*R*T**2*epsilon**3 + 0.00617283950617284*M*R*T**2*epsilon**2 - 0.00617283950617284*M*R*T**2*epsilon - 0.0185185185185185*M*S**3*epsilon**2 + 0.00925925925925926*M*S**3*epsilon - 0.037037037037037*M*S**2*T*epsilon**3 - 0.0246913580246914*M*S**2*T*epsilon**2 + 0.0246913580246914*M*S**2*T*epsilon - 0.00154320987654321*M*S**2*T - 0.0864197530864197*M*S*T**2*epsilon**3 + 0.0246913580246914*M*S*T**2*epsilon**2 + 0.0123456790123457*M*S*T**2*epsilon - 0.00154320987654321*M*S*T**2 - 0.0493827160493827*M*T**3*epsilon**3 + 0.0308641975308642*M*T**3*epsilon**2 - 0.00308641975308642*M*T**3*epsilon - 0.246913580246914*P**4*epsilon**3 + 0.123456790123457*P**4*epsilon**2 - 0.0493827160493827*P**3*R*epsilon**3 - 0.0246913580246914*P**3*R*epsilon**2 + 0.0246913580246914*P**3*R*epsilon + 0.246913580246914*P**3*S*epsilon**3 - 0.0617283950617284*P**3*S*epsilon + 0.493827160493827*P**3*T*epsilon**3 - 0.246913580246914*P**3*T*epsilon**2 + 0.0246913580246914*P**2*R*S*epsilon**3 + 0.037037037037037*P**2*R*S*epsilon**2 - 0.0185185185185185*P**2*R*S*epsilon - 0.00308641975308642*P**2*R*S + 0.0493827160493827*P**2*R*T*epsilon**3 + 0.0246913580246914*P**2*R*T*epsilon**2 - 0.0246913580246914*P**2*R*T*epsilon - 0.0617283950617284*P**2*S**2*epsilon**3 - 0.0802469135802469*P**2*S**2*epsilon**2 + 0.0432098765432099*P**2*S**2*epsilon + 0.00617283950617284*P**2*S**2 - 0.345679012345679*P**2*S*T*epsilon**3 + 0.0123456790123457*P**2*S*T*epsilon**2 + 0.0802469135802469*P**2*S*T*epsilon - 0.333333333333333*P**2*T**2*epsilon**3 + 0.166666666666667*P**2*T**2*epsilon**2 - 0.0123456790123457*P*R*S**2*epsilon**2 + 0.00308641975308642*P*R*S**2*epsilon + 0.00154320987654321*P*R*S**2 - 0.0123456790123457*P*R*S*T*epsilon**3 - 0.0185185185185185*P*R*S*T*epsilon**2 + 0.00925925925925926*P*R*S*T*epsilon + 0.00154320987654321*P*R*S*T - 0.0123456790123457*P*R*T**2*epsilon**3 - 0.00617283950617284*P*R*T**2*epsilon**2 + 0.00617283950617284*P*R*T**2*epsilon + 0.0246913580246914*P*S**3*epsilon**2 - 0.00308641975308642*P*S**3*epsilon - 0.00462962962962963*P*S**3 + 0.0493827160493827*P*S**2*T*epsilon**3 + 0.0617283950617284*P*S**2*T*epsilon**2 - 0.0339506172839506*P*S**2*T*epsilon - 0.00462962962962963*P*S**2*T + 0.135802469135802*P*S*T**2*epsilon**3 - 0.00617283950617284*P*S*T**2*epsilon**2 - 0.0308641975308642*P*S*T**2*epsilon + 0.0864197530864197*P*T**3*epsilon**3 - 0.0432098765432099*P*T**3*epsilon**2 - 0.00154320987654321*S**4*epsilon + 0.000771604938271605*S**4 - 0.00617283950617284*S**3*T*epsilon**2 + 0.00154320987654321*S**3*T - 0.00617283950617284*S**2*T**2*epsilon**3 - 0.00925925925925926*S**2*T**2*epsilon**2 + 0.00462962962962963*S**2*T**2*epsilon + 0.000771604938271605*S**2*T**2 - 0.0123456790123457*S*T**3*epsilon**3 + 0.00308641975308642*S*T**3*epsilon - 0.00617283950617284*T**4*epsilon**3 + 0.00308641975308642*T**4*epsilon**2) + UGA_star**3*(0.0617283950617284*M**3*P*epsilon**3 - 0.0740740740740741*M**3*P*epsilon**2 + 0.0277777777777778*M**3*P*epsilon - 0.00308641975308642*M**3*P + 0.0246913580246914*M**3*R*epsilon**3 - 0.0185185185185185*M**3*R*epsilon**2 + 0.00154320987654321*M**3*R - 0.037037037037037*M**3*S*epsilon**3 + 0.037037037037037*M**3*S*epsilon**2 - 0.00925925925925926*M**3*S*epsilon - 0.0493827160493827*M**3*T*epsilon**3 + 0.0555555555555556*M**3*T*epsilon**2 - 0.0185185185185185*M**3*T*epsilon + 0.00154320987654321*M**3*T - 0.358024691358025*M**2*P**2*epsilon**3 + 0.290123456790123*M**2*P**2*epsilon**2 - 0.058641975308642*M**2*P**2*epsilon + 0.00154320987654321*M**2*P**2 - 0.111111111111111*M**2*P*R*epsilon**3 + 0.0308641975308642*M**2*P*R*epsilon**2 + 0.0154320987654321*M**2*P*R*epsilon - 0.00154320987654321*M**2*P*R + 0.253086419753086*M**2*P*S*epsilon**3 - 0.114197530864198*M**2*P*S*epsilon**2 - 0.0200617283950617*M**2*P*S*epsilon + 0.00694444444444444*M**2*P*S + 0.425925925925926*M**2*P*T*epsilon**3 - 0.317901234567901*M**2*P*T*epsilon**2 + 0.0540123456790123*M**2*P*T*epsilon - 0.000771604938271605*M**2*P*T + 0.0123456790123457*M**2*R*S*epsilon**3 + 0.0185185185185185*M**2*R*S*epsilon**2 - 0.00925925925925926*M**2*R*S*epsilon - 0.00154320987654321*M**2*R*S + 0.037037037037037*M**2*R*T*epsilon**3 - 0.00925925925925926*M**2*R*T*epsilon - 0.0246913580246914*M**2*S**2*epsilon**3 - 0.0185185185185185*M**2*S**2*epsilon**2 + 0.0185185185185185*M**2*S**2*epsilon - 0.00154320987654321*M**2*S**2 - 0.123456790123457*M**2*S*T*epsilon**3 + 0.037037037037037*M**2*S*T*epsilon**2 + 0.0185185185185185*M**2*S*T*epsilon - 0.00308641975308642*M**2*S*T - 0.111111111111111*M**2*T**2*epsilon**3 + 0.0740740740740741*M**2*T**2*epsilon**2 - 0.00925925925925926*M**2*T**2*epsilon + 0.530864197530864*M*P**3*epsilon**3 - 0.320987654320988*M*P**3*epsilon**2 + 0.0277777777777778*M*P**3*epsilon + 0.135802469135802*M*P**2*R*epsilon**3 - 0.0246913580246914*M*P**2*R*epsilon**2 - 0.0216049382716049*M*P**2*R*epsilon - 0.419753086419753*M*P**2*S*epsilon**3 + 0.0925925925925926*M*P**2*S*epsilon**2 + 0.0648148148148148*M*P**2*S*epsilon - 0.00308641975308642*M*P**2*S - 0.740740740740741*M*P**2*T*epsilon**3 + 0.425925925925926*M*P**2*T*epsilon**2 - 0.0277777777777778*M*P**2*T*epsilon - 0.0246913580246914*M*P*R*S*epsilon**3 - 0.0246913580246914*M*P*R*S*epsilon**2 + 0.0154320987654321*M*P*R*S*epsilon + 0.00154320987654321*M*P*R*S - 0.0617283950617284*M*P*R*T*epsilon**3 + 0.00617283950617284*M*P*R*T*epsilon**2 + 0.0123456790123457*M*P*R*T*epsilon + 0.0617283950617284*M*P*S**2*epsilon**3 + 0.0555555555555556*M*P*S**2*epsilon**2 - 0.037037037037037*M*P*S**2*epsilon - 0.00308641975308642*M*P*S**2 + 0.308641975308642*M*P*S*T*epsilon**3 - 0.0432098765432099*M*P*S*T*epsilon**2 - 0.058641975308642*M*P*S*T*epsilon + 0.00154320987654321*M*P*S*T + 0.283950617283951*M*P*T**2*epsilon**3 - 0.154320987654321*M*P*T**2*epsilon**2 + 0.00617283950617284*M*P*T**2*epsilon - 0.00617283950617284*M*S**3*epsilon**2 + 0.00154320987654321*M*S**3 - 0.0123456790123457*M*S**2*T*epsilon**3 - 0.0185185185185185*M*S**2*T*epsilon**2 + 0.00925925925925926*M*S**2*T*epsilon + 0.00154320987654321*M*S**2*T - 0.037037037037037*M*S*T**2*epsilon**3 + 0.00925925925925926*M*S*T**2*epsilon - 0.0246913580246914*M*T**3*epsilon**3 + 0.0123456790123457*M*T**3*epsilon**2 - 0.222222222222222*P**4*epsilon**3 + 0.111111111111111*P**4*epsilon**2 - 0.0493827160493827*P**3*R*epsilon**3 + 0.0123456790123457*P**3*R*epsilon**2 + 0.00617283950617284*P**3*R*epsilon + 0.197530864197531*P**3*S*epsilon**3 - 0.0308641975308642*P**3*S*epsilon**2 - 0.0339506172839506*P**3*S*epsilon + 0.333333333333333*P**3*T*epsilon**3 - 0.166666666666667*P**3*T*epsilon**2 + 0.0123456790123457*P**2*R*S*epsilon**3 + 0.00617283950617284*P**2*R*S*epsilon**2 - 0.00617283950617284*P**2*R*S*epsilon + 0.0246913580246914*P**2*R*T*epsilon**3 - 0.00617283950617284*P**2*R*T*epsilon**2 - 0.00308641975308642*P**2*R*T*epsilon - 0.037037037037037*P**2*S**2*epsilon**3 - 0.0308641975308642*P**2*S**2*epsilon**2 + 0.0216049382716049*P**2*S**2*epsilon + 0.00154320987654321*P**2*S**2 - 0.172839506172839*P**2*S*T*epsilon**3 + 0.0246913580246914*P**2*S*T*epsilon**2 + 0.0308641975308642*P**2*S*T*epsilon - 0.148148148148148*P**2*T**2*epsilon**3 + 0.0740740740740741*P**2*T**2*epsilon**2 + 0.00617283950617284*P*S**3*epsilon**2 - 0.00154320987654321*P*S**3*epsilon - 0.000771604938271605*P*S**3 + 0.0123456790123457*P*S**2*T*epsilon**3 + 0.0123456790123457*P*S**2*T*epsilon**2 - 0.00771604938271605*P*S**2*T*epsilon - 0.000771604938271605*P*S**2*T + 0.0308641975308642*P*S*T**2*epsilon**3 - 0.00308641975308642*P*S*T**2*epsilon**2 - 0.00617283950617284*P*S*T**2*epsilon + 0.0185185185185185*P*T**3*epsilon**3 - 0.00925925925925926*P*T**3*epsilon**2) + UGA_star**2*UgA_star**2*(0.00617283950617284*M**3*P*epsilon**3 - 0.00925925925925926*M**3*P*epsilon**2 + 0.00462962962962963*M**3*P*epsilon - 0.000771604938271605*M**3*P + 0.00617283950617284*M**3*R*epsilon**3 - 0.00925925925925926*M**3*R*epsilon**2 + 0.00462962962962963*M**3*R*epsilon - 0.000771604938271605*M**3*R - 0.00617283950617284*M**3*S*epsilon**3 + 0.00925925925925926*M**3*S*epsilon**2 - 0.00462962962962963*M**3*S*epsilon + 0.000771604938271605*M**3*S - 0.00617283950617284*M**3*T*epsilon**3 + 0.00925925925925926*M**3*T*epsilon**2 - 0.00462962962962963*M**3*T*epsilon + 0.000771604938271605*M**3*T - 0.0740740740740741*M**2*P**2*epsilon**3 + 0.0679012345679012*M**2*P**2*epsilon**2 - 0.0169753086419753*M**2*P**2*epsilon + 0.000771604938271605*M**2*P**2 - 0.0802469135802469*M**2*P*R*epsilon**3 + 0.0648148148148148*M**2*P*R*epsilon**2 - 0.0138888888888889*M**2*P*R*epsilon + 0.000771604938271605*M**2*P*R + 0.0925925925925926*M**2*P*S*epsilon**3 - 0.0709876543209877*M**2*P*S*epsilon**2 + 0.0108024691358025*M**2*P*S*epsilon + 0.000771604938271605*M**2*P*S + 0.117283950617284*M**2*P*T*epsilon**3 - 0.101851851851852*M**2*P*T*epsilon**2 + 0.0231481481481481*M**2*P*T*epsilon - 0.000771604938271605*M**2*P*T - 0.00617283950617284*M**2*R**2*epsilon**3 - 0.00308641975308642*M**2*R**2*epsilon**2 + 0.00308641975308642*M**2*R**2*epsilon + 0.0246913580246914*M**2*R*S*epsilon**3 - 0.00925925925925926*M**2*R*S*epsilon + 0.00154320987654321*M**2*R*S + 0.0493827160493827*M**2*R*T*epsilon**3 - 0.0308641975308642*M**2*R*T*epsilon**2 + 0.00308641975308642*M**2*R*T*epsilon - 0.0185185185185185*M**2*S**2*epsilon**3 + 0.00308641975308642*M**2*S**2*epsilon**2 + 0.00617283950617284*M**2*S**2*epsilon - 0.00154320987654321*M**2*S**2 - 0.0617283950617284*M**2*S*T*epsilon**3 + 0.037037037037037*M**2*S*T*epsilon**2 - 0.00154320987654321*M**2*S*T - 0.0432098765432099*M**2*T**2*epsilon**3 + 0.0339506172839506*M**2*T**2*epsilon**2 - 0.00617283950617284*M**2*T**2*epsilon + 0.160493827160494*M*P**3*epsilon**3 - 0.104938271604938*M*P**3*epsilon**2 + 0.0123456790123457*M*P**3*epsilon + 0.172839506172839*M*P**2*R*epsilon**3 - 0.0987654320987654*M*P**2*R*epsilon**2 + 0.00617283950617284*M*P**2*R*epsilon - 0.209876543209877*M*P**2*S*epsilon**3 + 0.0925925925925926*M*P**2*S*epsilon**2 + 0.00925925925925926*M*P**2*S*epsilon - 0.00154320987654321*M*P**2*S - 0.296296296296296*M*P**2*T*epsilon**3 + 0.185185185185185*M*P**2*T*epsilon**2 - 0.0185185185185185*M*P**2*T*epsilon + 0.0123456790123457*M*P*R**2*epsilon**3 + 0.00617283950617284*M*P*R**2*epsilon**2 - 0.00617283950617284*M*P*R**2*epsilon - 0.0617283950617284*M*P*R*S*epsilon**3 - 0.0185185185185185*M*P*R*S*epsilon**2 + 0.0277777777777778*M*P*R*S*epsilon - 0.00154320987654321*M*P*R*S - 0.148148148148148*M*P*R*T*epsilon**3 + 0.0740740740740741*M*P*R*T*epsilon**2 + 0.0493827160493827*M*P*S**2*epsilon**3 + 0.0185185185185185*M*P*S**2*epsilon**2 - 0.0231481481481481*M*P*S**2*epsilon + 0.000771604938271605*M*P*S**2 + 0.197530864197531*M*P*S*T*epsilon**3 - 0.0617283950617284*M*P*S*T*epsilon**2 - 0.0216049382716049*M*P*S*T*epsilon + 0.00154320987654321*M*P*S*T + 0.154320987654321*M*P*T**2*epsilon**3 - 0.0895061728395062*M*P*T**2*epsilon**2 + 0.00617283950617284*M*P*T**2*epsilon + 0.00617283950617284*M*R*S**2*epsilon**2 - 0.00154320987654321*M*R*S**2*epsilon - 0.000771604938271605*M*R*S**2 + 0.0123456790123457*M*R*S*T*epsilon**3 + 0.00617283950617284*M*R*S*T*epsilon**2 - 0.00617283950617284*M*R*S*T*epsilon + 0.0185185185185185*M*R*T**2*epsilon**3 - 0.00925925925925926*M*R*T**2*epsilon**2 - 0.00617283950617284*M*S**3*epsilon**2 + 0.00154320987654321*M*S**3*epsilon + 0.000771604938271605*M*S**3 - 0.0123456790123457*M*S**2*T*epsilon**3 - 0.0123456790123457*M*S**2*T*epsilon**2 + 0.00771604938271605*M*S**2*T*epsilon + 0.000771604938271605*M*S**2*T - 0.0308641975308642*M*S*T**2*epsilon**3 + 0.00308641975308642*M*S*T**2*epsilon**2 + 0.00617283950617284*M*S*T**2*epsilon - 0.0185185185185185*M*T**3*epsilon**3 + 0.00925925925925926*M*T**3*epsilon**2 - 0.0925925925925926*P**4*epsilon**3 + 0.0462962962962963*P**4*epsilon**2 - 0.0987654320987654*P**3*R*epsilon**3 + 0.0432098765432099*P**3*R*epsilon**2 + 0.00308641975308642*P**3*R*epsilon + 0.123456790123457*P**3*S*epsilon**3 - 0.0308641975308642*P**3*S*epsilon**2 - 0.0154320987654321*P**3*S*epsilon + 0.185185185185185*P**3*T*epsilon**3 - 0.0925925925925926*P**3*T*epsilon**2 - 0.00617283950617284*P**2*R**2*epsilon**3 - 0.00308641975308642*P**2*R**2*epsilon**2 + 0.00308641975308642*P**2*R**2*epsilon + 0.037037037037037*P**2*R*S*epsilon**3 + 0.0185185185185185*P**2*R*S*epsilon**2 - 0.0185185185185185*P**2*R*S*epsilon + 0.0987654320987654*P**2*R*T*epsilon**3 - 0.0432098765432099*P**2*R*T*epsilon**2 - 0.00308641975308642*P**2*R*T*epsilon - 0.0308641975308642*P**2*S**2*epsilon**3 - 0.0216049382716049*P**2*S**2*epsilon**2 + 0.0169753086419753*P**2*S**2*epsilon + 0.000771604938271605*P**2*S**2 - 0.135802469135802*P**2*S*T*epsilon**3 + 0.0246913580246914*P**2*S*T*epsilon**2 + 0.0216049382716049*P**2*S*T*epsilon - 0.111111111111111*P**2*T**2*epsilon**3 + 0.0555555555555556*P**2*T**2*epsilon**2 - 0.00617283950617284*P*R*S**2*epsilon**2 + 0.00154320987654321*P*R*S**2*epsilon + 0.000771604938271605*P*R*S**2 - 0.0123456790123457*P*R*S*T*epsilon**3 - 0.00617283950617284*P*R*S*T*epsilon**2 + 0.00617283950617284*P*R*S*T*epsilon - 0.0185185185185185*P*R*T**2*epsilon**3 + 0.00925925925925926*P*R*T**2*epsilon**2 + 0.00617283950617284*P*S**3*epsilon**2 - 0.00154320987654321*P*S**3*epsilon - 0.000771604938271605*P*S**3 + 0.0123456790123457*P*S**2*T*epsilon**3 + 0.0123456790123457*P*S**2*T*epsilon**2 - 0.00771604938271605*P*S**2*T*epsilon - 0.000771604938271605*P*S**2*T + 0.0308641975308642*P*S*T**2*epsilon**3 - 0.00308641975308642*P*S*T**2*epsilon**2 - 0.00617283950617284*P*S*T**2*epsilon + 0.0185185185185185*P*T**3*epsilon**3 - 0.00925925925925926*P*T**3*epsilon**2) + UGA_star**2*UgA_star*(-0.0123456790123457*M**3*P*epsilon**3 + 0.0185185185185185*M**3*P*epsilon**2 - 0.00925925925925926*M**3*P*epsilon + 0.00154320987654321*M**3*P + 0.00617283950617284*M**3*S*epsilon**3 - 0.00925925925925926*M**3*S*epsilon**2 + 0.00462962962962963*M**3*S*epsilon - 0.000771604938271605*M**3*S + 0.00617283950617284*M**3*T*epsilon**3 - 0.00925925925925926*M**3*T*epsilon**2 + 0.00462962962962963*M**3*T*epsilon - 0.000771604938271605*M**3*T + 0.148148148148148*M**2*P**2*epsilon**3 - 0.135802469135802*M**2*P**2*epsilon**2 + 0.0339506172839506*M**2*P**2*epsilon - 0.00154320987654321*M**2*P**2 + 0.0123456790123457*M**2*P*R*epsilon**3 + 0.00617283950617284*M**2*P*R*epsilon**2 - 0.00617283950617284*M**2*P*R*epsilon - 0.111111111111111*M**2*P*S*epsilon**3 + 0.0740740740740741*M**2*P*S*epsilon**2 - 0.00462962962962963*M**2*P*S*epsilon - 0.00231481481481481*M**2*P*S - 0.160493827160494*M**2*P*T*epsilon**3 + 0.135802469135802*M**2*P*T*epsilon**2 - 0.029320987654321*M**2*P*T*epsilon + 0.000771604938271605*M**2*P*T - 0.00617283950617284*M**2*R*S*epsilon**3 - 0.00308641975308642*M**2*R*S*epsilon**2 + 0.00308641975308642*M**2*R*S*epsilon - 0.00617283950617284*M**2*R*T*epsilon**3 - 0.00308641975308642*M**2*R*T*epsilon**2 + 0.00308641975308642*M**2*R*T*epsilon + 0.0185185185185185*M**2*S**2*epsilon**3 - 0.00308641975308642*M**2*S**2*epsilon**2 - 0.00617283950617284*M**2*S**2*epsilon + 0.00154320987654321*M**2*S**2 + 0.0617283950617284*M**2*S*T*epsilon**3 - 0.037037037037037*M**2*S*T*epsilon**2 + 0.00154320987654321*M**2*S*T + 0.0432098765432099*M**2*T**2*epsilon**3 - 0.0339506172839506*M**2*T**2*epsilon**2 + 0.00617283950617284*M**2*T**2*epsilon - 0.320987654320988*M*P**3*epsilon**3 + 0.209876543209877*M*P**3*epsilon**2 - 0.0246913580246914*M*P**3*epsilon - 0.0246913580246914*M*P**2*R*epsilon**3 - 0.0123456790123457*M*P**2*R*epsilon**2 + 0.0123456790123457*M*P**2*R*epsilon + 0.259259259259259*M*P**2*S*epsilon**3 - 0.0802469135802469*M*P**2*S*epsilon**2 - 0.0308641975308642*M*P**2*S*epsilon + 0.00308641975308642*M*P**2*S + 0.432098765432099*M*P**2*T*epsilon**3 - 0.265432098765432*M*P**2*T*epsilon**2 + 0.0246913580246914*M*P**2*T*epsilon + 0.0123456790123457*M*P*R*S*epsilon**3 + 0.00617283950617284*M*P*R*S*epsilon**2 - 0.00617283950617284*M*P*R*S*epsilon + 0.0123456790123457*M*P*R*T*epsilon**3 + 0.00617283950617284*M*P*R*T*epsilon**2 - 0.00617283950617284*M*P*R*T*epsilon - 0.0493827160493827*M*P*S**2*epsilon**3 - 0.0246913580246914*M*P*S**2*epsilon**2 + 0.0246913580246914*M*P*S**2*epsilon - 0.209876543209877*M*P*S*T*epsilon**3 + 0.0555555555555556*M*P*S*T*epsilon**2 + 0.0277777777777778*M*P*S*T*epsilon - 0.00154320987654321*M*P*S*T - 0.172839506172839*M*P*T**2*epsilon**3 + 0.0987654320987654*M*P*T**2*epsilon**2 - 0.00617283950617284*M*P*T**2*epsilon + 0.00617283950617284*M*S**3*epsilon**2 - 0.00154320987654321*M*S**3*epsilon - 0.000771604938271605*M*S**3 + 0.0123456790123457*M*S**2*T*epsilon**3 + 0.0123456790123457*M*S**2*T*epsilon**2 - 0.00771604938271605*M*S**2*T*epsilon - 0.000771604938271605*M*S**2*T + 0.0308641975308642*M*S*T**2*epsilon**3 - 0.00308641975308642*M*S*T**2*epsilon**2 - 0.00617283950617284*M*S*T**2*epsilon + 0.0185185185185185*M*T**3*epsilon**3 - 0.00925925925925926*M*T**3*epsilon**2 + 0.185185185185185*P**4*epsilon**3 - 0.0925925925925926*P**4*epsilon**2 + 0.0123456790123457*P**3*R*epsilon**3 + 0.00617283950617284*P**3*R*epsilon**2 - 0.00617283950617284*P**3*R*epsilon - 0.154320987654321*P**3*S*epsilon**3 + 0.0154320987654321*P**3*S*epsilon**2 + 0.0308641975308642*P**3*S*epsilon - 0.277777777777778*P**3*T*epsilon**3 + 0.138888888888889*P**3*T*epsilon**2 - 0.00617283950617284*P**2*R*S*epsilon**3 - 0.00308641975308642*P**2*R*S*epsilon**2 + 0.00308641975308642*P**2*R*S*epsilon - 0.00617283950617284*P**2*R*T*epsilon**3 - 0.00308641975308642*P**2*R*T*epsilon**2 + 0.00308641975308642*P**2*R*T*epsilon + 0.0308641975308642*P**2*S**2*epsilon**3 + 0.0277777777777778*P**2*S**2*epsilon**2 - 0.0185185185185185*P**2*S**2*epsilon - 0.00154320987654321*P**2*S**2 + 0.148148148148148*P**2*S*T*epsilon**3 - 0.0185185185185185*P**2*S*T*epsilon**2 - 0.0277777777777778*P**2*S*T*epsilon + 0.12962962962963*P**2*T**2*epsilon**3 - 0.0648148148148148*P**2*T**2*epsilon**2 - 0.00617283950617284*P*S**3*epsilon**2 + 0.00154320987654321*P*S**3*epsilon + 0.000771604938271605*P*S**3 - 0.0123456790123457*P*S**2*T*epsilon**3 - 0.0123456790123457*P*S**2*T*epsilon**2 + 0.00771604938271605*P*S**2*T*epsilon + 0.000771604938271605*P*S**2*T - 0.0308641975308642*P*S*T**2*epsilon**3 + 0.00308641975308642*P*S*T**2*epsilon**2 + 0.00617283950617284*P*S*T**2*epsilon - 0.0185185185185185*P*T**3*epsilon**3 + 0.00925925925925926*P*T**3*epsilon**2) + UGA_star**2*(0.00617283950617284*M**4*epsilon**3 - 0.00925925925925926*M**4*epsilon**2 + 0.00462962962962963*M**4*epsilon - 0.000771604938271605*M**4 - 0.104938271604938*M**3*P*epsilon**3 + 0.095679012345679*M**3*P*epsilon**2 - 0.0231481481481481*M**3*P*epsilon + 0.000771604938271605*M**3*P - 0.0123456790123457*M**3*R*epsilon**3 + 0.00308641975308642*M**3*R*epsilon + 0.037037037037037*M**3*S*epsilon**3 - 0.0185185185185185*M**3*S*epsilon**2 - 0.00308641975308642*M**3*S*epsilon + 0.00154320987654321*M**3*S + 0.0617283950617284*M**3*T*epsilon**3 - 0.0493827160493827*M**3*T*epsilon**2 + 0.00925925925925926*M**3*T*epsilon + 0.302469135802469*M**2*P**2*epsilon**3 - 0.200617283950617*M**2*P**2*epsilon**2 + 0.0246913580246914*M**2*P**2*epsilon + 0.0308641975308642*M**2*P*R*epsilon**3 - 0.00308641975308642*M**2*P*R*epsilon**2 - 0.00617283950617284*M**2*P*R*epsilon - 0.141975308641975*M**2*P*S*epsilon**3 + 0.0339506172839506*M**2*P*S*epsilon**2 + 0.0216049382716049*M**2*P*S*epsilon - 0.00154320987654321*M**2*P*S - 0.253086419753086*M**2*P*T*epsilon**3 + 0.151234567901235*M**2*P*T*epsilon**2 - 0.0123456790123457*M**2*P*T*epsilon + 0.00617283950617284*M**2*S**2*epsilon**3 + 0.00925925925925926*M**2*S**2*epsilon**2 - 0.00462962962962963*M**2*S**2*epsilon - 0.000771604938271605*M**2*S**2 + 0.037037037037037*M**2*S*T*epsilon**3 - 0.00925925925925926*M**2*S*T*epsilon + 0.037037037037037*M**2*T**2*epsilon**3 - 0.0185185185185185*M**2*T**2*epsilon**2 - 0.283950617283951*M*P**3*epsilon**3 + 0.154320987654321*M*P**3*epsilon**2 - 0.00617283950617284*M*P**3*epsilon - 0.0246913580246914*M*P**2*R*epsilon**3 + 0.00617283950617284*M*P**2*R*epsilon**2 + 0.00308641975308642*M*P**2*R*epsilon + 0.160493827160494*M*P**2*S*epsilon**3 - 0.0308641975308642*M*P**2*S*epsilon**2 - 0.0246913580246914*M*P**2*S*epsilon + 0.271604938271605*M*P**2*T*epsilon**3 - 0.141975308641975*M*P**2*T*epsilon**2 + 0.00308641975308642*M*P**2*T*epsilon - 0.0123456790123457*M*P*S**2*epsilon**3 - 0.0123456790123457*M*P*S**2*epsilon**2 + 0.00771604938271605*M*P*S**2*epsilon + 0.000771604938271605*M*P*S**2 - 0.0617283950617284*M*P*S*T*epsilon**3 + 0.00617283950617284*M*P*S*T*epsilon**2 + 0.0123456790123457*M*P*S*T*epsilon - 0.0555555555555556*M*P*T**2*epsilon**3 + 0.0277777777777778*M*P*T**2*epsilon**2 + 0.0802469135802469*P**4*epsilon**3 - 0.0401234567901235*P**4*epsilon**2 + 0.00617283950617284*P**3*R*epsilon**3 - 0.00308641975308642*P**3*R*epsilon**2 - 0.0555555555555556*P**3*S*epsilon**3 + 0.0154320987654321*P**3*S*epsilon**2 + 0.00617283950617284*P**3*S*epsilon - 0.0802469135802469*P**3*T*epsilon**3 + 0.0401234567901235*P**3*T*epsilon**2 + 0.00617283950617284*P**2*S**2*epsilon**3 + 0.00308641975308642*P**2*S**2*epsilon**2 - 0.00308641975308642*P**2*S**2*epsilon + 0.0246913580246914*P**2*S*T*epsilon**3 - 0.00617283950617284*P**2*S*T*epsilon**2 - 0.00308641975308642*P**2*S*T*epsilon + 0.0185185185185185*P**2*T**2*epsilon**3 - 0.00925925925925926*P**2*T**2*epsilon**2) + UGA_star*UgA_star**2*(-0.0123456790123457*M**3*P*epsilon**3 + 0.0123456790123457*M**3*P*epsilon**2 - 0.00308641975308642*M**3*P*epsilon - 0.0123456790123457*M**3*R*epsilon**3 + 0.0123456790123457*M**3*R*epsilon**2 - 0.00308641975308642*M**3*R*epsilon + 0.0123456790123457*M**3*S*epsilon**3 - 0.0123456790123457*M**3*S*epsilon**2 + 0.00308641975308642*M**3*S*epsilon + 0.0123456790123457*M**3*T*epsilon**3 - 0.0123456790123457*M**3*T*epsilon**2 + 0.00308641975308642*M**3*T*epsilon + 0.0617283950617284*M**2*P**2*epsilon**3 - 0.0432098765432099*M**2*P**2*epsilon**2 + 0.00617283950617284*M**2*P**2*epsilon + 0.0617283950617284*M**2*P*R*epsilon**3 - 0.0432098765432099*M**2*P*R*epsilon**2 + 0.00617283950617284*M**2*P*R*epsilon - 0.0679012345679012*M**2*P*S*epsilon**3 + 0.0401234567901235*M**2*P*S*epsilon**2 - 0.00308641975308642*M**2*P*S*epsilon - 0.0802469135802469*M**2*P*T*epsilon**3 + 0.0524691358024691*M**2*P*T*epsilon**2 - 0.00617283950617284*M**2*P*T*epsilon - 0.00617283950617284*M**2*R*S*epsilon**3 - 0.00308641975308642*M**2*R*S*epsilon**2 + 0.00308641975308642*M**2*R*S*epsilon - 0.0185185185185185*M**2*R*T*epsilon**3 + 0.00925925925925926*M**2*R*T*epsilon**2 + 0.00617283950617284*M**2*S**2*epsilon**3 + 0.00308641975308642*M**2*S**2*epsilon**2 - 0.00308641975308642*M**2*S**2*epsilon + 0.0246913580246914*M**2*S*T*epsilon**3 - 0.00617283950617284*M**2*S*T*epsilon**2 - 0.00308641975308642*M**2*S*T*epsilon + 0.0185185185185185*M**2*T**2*epsilon**3 - 0.00925925925925926*M**2*T**2*epsilon**2 - 0.0864197530864197*M*P**3*epsilon**3 + 0.0493827160493827*M*P**3*epsilon**2 - 0.00308641975308642*M*P**3*epsilon - 0.0864197530864197*M*P**2*R*epsilon**3 + 0.0493827160493827*M*P**2*R*epsilon**2 - 0.00308641975308642*M*P**2*R*epsilon + 0.0987654320987654*M*P**2*S*epsilon**3 - 0.0432098765432099*M*P**2*S*epsilon**2 - 0.00308641975308642*M*P**2*S*epsilon + 0.123456790123457*M*P**2*T*epsilon**3 - 0.0679012345679012*M*P**2*T*epsilon**2 + 0.00308641975308642*M*P**2*T*epsilon + 0.0123456790123457*M*P*R*S*epsilon**3 + 0.00617283950617284*M*P*R*S*epsilon**2 - 0.00617283950617284*M*P*R*S*epsilon + 0.037037037037037*M*P*R*T*epsilon**3 - 0.0185185185185185*M*P*R*T*epsilon**2 - 0.0123456790123457*M*P*S**2*epsilon**3 - 0.00617283950617284*M*P*S**2*epsilon**2 + 0.00617283950617284*M*P*S**2*epsilon - 0.0493827160493827*M*P*S*T*epsilon**3 + 0.0123456790123457*M*P*S*T*epsilon**2 + 0.00617283950617284*M*P*S*T*epsilon - 0.037037037037037*M*P*T**2*epsilon**3 + 0.0185185185185185*M*P*T**2*epsilon**2 + 0.037037037037037*P**4*epsilon**3 - 0.0185185185185185*P**4*epsilon**2 + 0.037037037037037*P**3*R*epsilon**3 - 0.0185185185185185*P**3*R*epsilon**2 - 0.0432098765432099*P**3*S*epsilon**3 + 0.0154320987654321*P**3*S*epsilon**2 + 0.00308641975308642*P**3*S*epsilon - 0.0555555555555556*P**3*T*epsilon**3 + 0.0277777777777778*P**3*T*epsilon**2 - 0.00617283950617284*P**2*R*S*epsilon**3 - 0.00308641975308642*P**2*R*S*epsilon**2 + 0.00308641975308642*P**2*R*S*epsilon - 0.0185185185185185*P**2*R*T*epsilon**3 + 0.00925925925925926*P**2*R*T*epsilon**2 + 0.00617283950617284*P**2*S**2*epsilon**3 + 0.00308641975308642*P**2*S**2*epsilon**2 - 0.00308641975308642*P**2*S**2*epsilon + 0.0246913580246914*P**2*S*T*epsilon**3 - 0.00617283950617284*P**2*S*T*epsilon**2 - 0.00308641975308642*P**2*S*T*epsilon + 0.0185185185185185*P**2*T**2*epsilon**3 - 0.00925925925925926*P**2*T**2*epsilon**2) + UGA_star*UgA_star*(0.0246913580246914*M**3*P*epsilon**3 - 0.0246913580246914*M**3*P*epsilon**2 + 0.00617283950617284*M**3*P*epsilon - 0.0123456790123457*M**3*S*epsilon**3 + 0.0123456790123457*M**3*S*epsilon**2 - 0.00308641975308642*M**3*S*epsilon - 0.0123456790123457*M**3*T*epsilon**3 + 0.0123456790123457*M**3*T*epsilon**2 - 0.00308641975308642*M**3*T*epsilon - 0.123456790123457*M**2*P**2*epsilon**3 + 0.0864197530864197*M**2*P**2*epsilon**2 - 0.0123456790123457*M**2*P**2*epsilon + 0.0740740740740741*M**2*P*S*epsilon**3 - 0.037037037037037*M**2*P*S*epsilon**2 + 0.0987654320987654*M**2*P*T*epsilon**3 - 0.0617283950617284*M**2*P*T*epsilon**2 + 0.00617283950617284*M**2*P*T*epsilon - 0.00617283950617284*M**2*S**2*epsilon**3 - 0.00308641975308642*M**2*S**2*epsilon**2 + 0.00308641975308642*M**2*S**2*epsilon - 0.0246913580246914*M**2*S*T*epsilon**3 + 0.00617283950617284*M**2*S*T*epsilon**2 + 0.00308641975308642*M**2*S*T*epsilon - 0.0185185185185185*M**2*T**2*epsilon**3 + 0.00925925925925926*M**2*T**2*epsilon**2 + 0.172839506172839*M*P**3*epsilon**3 - 0.0987654320987654*M*P**3*epsilon**2 + 0.00617283950617284*M*P**3*epsilon - 0.111111111111111*M*P**2*S*epsilon**3 + 0.037037037037037*M*P**2*S*epsilon**2 + 0.00925925925925926*M*P**2*S*epsilon - 0.160493827160494*M*P**2*T*epsilon**3 + 0.0864197530864197*M*P**2*T*epsilon**2 - 0.00308641975308642*M*P**2*T*epsilon + 0.0123456790123457*M*P*S**2*epsilon**3 + 0.00617283950617284*M*P*S**2*epsilon**2 - 0.00617283950617284*M*P*S**2*epsilon + 0.0493827160493827*M*P*S*T*epsilon**3 - 0.0123456790123457*M*P*S*T*epsilon**2 - 0.00617283950617284*M*P*S*T*epsilon + 0.037037037037037*M*P*T**2*epsilon**3 - 0.0185185185185185*M*P*T**2*epsilon**2 - 0.0740740740740741*P**4*epsilon**3 + 0.037037037037037*P**4*epsilon**2 + 0.0493827160493827*P**3*S*epsilon**3 - 0.0123456790123457*P**3*S*epsilon**2 - 0.00617283950617284*P**3*S*epsilon + 0.0740740740740741*P**3*T*epsilon**3 - 0.037037037037037*P**3*T*epsilon**2 - 0.00617283950617284*P**2*S**2*epsilon**3 - 0.00308641975308642*P**2*S**2*epsilon**2 + 0.00308641975308642*P**2*S**2*epsilon - 0.0246913580246914*P**2*S*T*epsilon**3 + 0.00617283950617284*P**2*S*T*epsilon**2 + 0.00308641975308642*P**2*S*T*epsilon - 0.0185185185185185*P**2*T**2*epsilon**3 + 0.00925925925925926*P**2*T**2*epsilon**2) + UGA_star*(-0.0123456790123457*M**4*epsilon**3 + 0.0123456790123457*M**4*epsilon**2 - 0.00308641975308642*M**4*epsilon + 0.0740740740740741*M**3*P*epsilon**3 - 0.0493827160493827*M**3*P*epsilon**2 + 0.00617283950617284*M**3*P*epsilon - 0.0123456790123457*M**3*S*epsilon**3 + 0.00308641975308642*M**3*S*epsilon - 0.0246913580246914*M**3*T*epsilon**3 + 0.0123456790123457*M**3*T*epsilon**2 - 0.123456790123457*M**2*P**2*epsilon**3 + 0.0679012345679012*M**2*P**2*epsilon**2 - 0.00308641975308642*M**2*P**2*epsilon + 0.0308641975308642*M**2*P*S*epsilon**3 - 0.00308641975308642*M**2*P*S*epsilon**2 - 0.00617283950617284*M**2*P*S*epsilon + 0.0555555555555556*M**2*P*T*epsilon**3 - 0.0277777777777778*M**2*P*T*epsilon**2 + 0.0740740740740741*M*P**3*epsilon**3 - 0.037037037037037*M*P**3*epsilon**2 - 0.0246913580246914*M*P**2*S*epsilon**3 + 0.00617283950617284*M*P**2*S*epsilon**2 + 0.00308641975308642*M*P**2*S*epsilon - 0.037037037037037*M*P**2*T*epsilon**3 + 0.0185185185185185*M*P**2*T*epsilon**2 - 0.0123456790123457*P**4*epsilon**3 + 0.00617283950617284*P**4*epsilon**2 + 0.00617283950617284*P**3*S*epsilon**3 - 0.00308641975308642*P**3*S*epsilon**2 + 0.00617283950617284*P**3*T*epsilon**3 - 0.00308641975308642*P**3*T*epsilon**2) + UgA_star**2*(0.00617283950617284*M**3*P*epsilon**3 - 0.00308641975308642*M**3*P*epsilon**2 + 0.00617283950617284*M**3*R*epsilon**3 - 0.00308641975308642*M**3*R*epsilon**2 - 0.00617283950617284*M**3*S*epsilon**3 + 0.00308641975308642*M**3*S*epsilon**2 - 0.00617283950617284*M**3*T*epsilon**3 + 0.00308641975308642*M**3*T*epsilon**2 - 0.0185185185185185*M**2*P**2*epsilon**3 + 0.00925925925925926*M**2*P**2*epsilon**2 - 0.0185185185185185*M**2*P*R*epsilon**3 + 0.00925925925925926*M**2*P*R*epsilon**2 + 0.0185185185185185*M**2*P*S*epsilon**3 - 0.00925925925925926*M**2*P*S*epsilon**2 + 0.0185185185185185*M**2*P*T*epsilon**3 - 0.00925925925925926*M**2*P*T*epsilon**2 + 0.0185185185185185*M*P**3*epsilon**3 - 0.00925925925925926*M*P**3*epsilon**2 + 0.0185185185185185*M*P**2*R*epsilon**3 - 0.00925925925925926*M*P**2*R*epsilon**2 - 0.0185185185185185*M*P**2*S*epsilon**3 + 0.00925925925925926*M*P**2*S*epsilon**2 - 0.0185185185185185*M*P**2*T*epsilon**3 + 0.00925925925925926*M*P**2*T*epsilon**2 - 0.00617283950617284*P**4*epsilon**3 + 0.00308641975308642*P**4*epsilon**2 - 0.00617283950617284*P**3*R*epsilon**3 + 0.00308641975308642*P**3*R*epsilon**2 + 0.00617283950617284*P**3*S*epsilon**3 - 0.00308641975308642*P**3*S*epsilon**2 + 0.00617283950617284*P**3*T*epsilon**3 - 0.00308641975308642*P**3*T*epsilon**2) + UgA_star*(-0.0123456790123457*M**3*P*epsilon**3 + 0.00617283950617284*M**3*P*epsilon**2 + 0.00617283950617284*M**3*S*epsilon**3 - 0.00308641975308642*M**3*S*epsilon**2 + 0.00617283950617284*M**3*T*epsilon**3 - 0.00308641975308642*M**3*T*epsilon**2 + 0.037037037037037*M**2*P**2*epsilon**3 - 0.0185185185185185*M**2*P**2*epsilon**2 - 0.0185185185185185*M**2*P*S*epsilon**3 + 0.00925925925925926*M**2*P*S*epsilon**2 - 0.0185185185185185*M**2*P*T*epsilon**3 + 0.00925925925925926*M**2*P*T*epsilon**2 - 0.037037037037037*M*P**3*epsilon**3 + 0.0185185185185185*M*P**3*epsilon**2 + 0.0185185185185185*M*P**2*S*epsilon**3 - 0.00925925925925926*M*P**2*S*epsilon**2 + 0.0185185185185185*M*P**2*T*epsilon**3 - 0.00925925925925926*M*P**2*T*epsilon**2 + 0.0123456790123457*P**4*epsilon**3 - 0.00617283950617284*P**4*epsilon**2 - 0.00617283950617284*P**3*S*epsilon**3 + 0.00308641975308642*P**3*S*epsilon**2 - 0.00617283950617284*P**3*T*epsilon**3 + 0.00308641975308642*P**3*T*epsilon**2))"
      ]
     },
     "execution_count": 72,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(sym.expand(n), N_G, UGA_star, UgA_star)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "N_G**2*(-512.0*epsilon**2 - 512.0*epsilon - 128.0)*(-0.5*M + 0.5*P + UGA_star**2*(0.5*P + 0.5*R - 0.5*S - 0.5*T) + UGA_star*(-1.0*P + 0.5*S + 0.5*T))**2*(-0.5*M*epsilon - 0.25*N_G*epsilon - 0.125*N_G + 0.5*P*epsilon + UGA_star**2*(0.5*P*epsilon + 0.25*R - 0.25*S - 0.5*T*epsilon) + UGA_star*(0.5*M*epsilon - 0.25*M - 1.0*P*epsilon + 0.25*S + 0.5*T*epsilon))"
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.factor(sym.expand(d), UGA_star)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-56-ffaacff58f58>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mresult\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msym\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfactor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msym\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0msym\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mtogether\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mevaluated_F_jac\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mUGA_star\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mUgA_star\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/function.py\u001b[0m in \u001b[0;36mexpand\u001b[0;34m(e, deep, modulus, power_base, power_exp, mul, log, multinomial, basic, **hints)\u001b[0m\n\u001b[1;32m   2321\u001b[0m     \u001b[0mhints\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m'multinomial'\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mmultinomial\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2322\u001b[0m     \u001b[0mhints\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;34m'basic'\u001b[0m\u001b[0;34m]\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mbasic\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 2323\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0msympify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0me\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mexpand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mdeep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmodulus\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mmodulus\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   2324\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   2325\u001b[0m \u001b[0;31m# This is a special application of two hints\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mexpand\u001b[0;34m(self, deep, modulus, power_base, power_exp, mul, log, multinomial, basic, **hints)\u001b[0m\n\u001b[1;32m   3122\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0muse_hint\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3123\u001b[0m                 \u001b[0mhint\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m'_eval_expand_'\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3124\u001b[0;31m                 \u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mdeep\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3125\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3126\u001b[0m         \u001b[0;32mwhile\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3055\u001b[0m             \u001b[0msargs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3056\u001b[0m             \u001b[0;32mfor\u001b[0m \u001b[0marg\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3057\u001b[0;31m                 \u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0marghit\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_expand_hint\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3058\u001b[0m                 \u001b[0mhit\u001b[0m \u001b[0;34m|=\u001b[0m \u001b[0marghit\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3059\u001b[0m                 \u001b[0msargs\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0marg\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36m_expand_hint\u001b[0;34m(expr, hint, deep, **hints)\u001b[0m\n\u001b[1;32m   3063\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3064\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mhasattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3065\u001b[0;31m             \u001b[0mnewexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mgetattr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mhint\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3066\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mnewexpr\u001b[0m \u001b[0;34m!=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3067\u001b[0m                 \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mnewexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/power.py\u001b[0m in \u001b[0;36m_eval_expand_power_base\u001b[0;34m(self, **hints)\u001b[0m\n\u001b[1;32m    834\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mfunc\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    835\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 836\u001b[0;31m     \u001b[0;32mdef\u001b[0m \u001b[0m_eval_expand_power_base\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mhints\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    837\u001b[0m         \u001b[0;34m\"\"\"(a*b)**n -> a**n * b**n\"\"\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    838\u001b[0m         \u001b[0mforce\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mhints\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mget\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'force'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mFalse\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "result = sym.factor(sym.expand(sym.together(evaluated_F_jac[0,0]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Finished with element 0,0!\n",
      "Finished with element 0,1!\n",
      "Finished with element 0,2!\n",
      "Finished with element 1,0!\n",
      "Finished with element 1,1!\n",
      "Finished with element 1,2!\n",
      "Finished with element 2,0!\n",
      "Finished with element 2,1!\n",
      "Finished with element 2,2!\n"
     ]
    }
   ],
   "source": [
    "simplified_F_jac = sym.zeros(3, 3)\n",
    "\n",
    "for i in range(3):\n",
    "    for j in range(3):\n",
    "        simplified_F_jac[i, j] = sym.factor(sym.cancel(sym.together(evaluated_F_jac[i, j])))\n",
    "        print(\"Finished with element {},{}!\".format(i,j))\n",
    "        "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 57,
   "metadata": {},
   "outputs": [
    {
     "ename": "KeyboardInterrupt",
     "evalue": "",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m                         Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-57-e9ab42944c6b>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0meigenvals\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msimplified_F_jac\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0meigenvals\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/matrices/matrices.py\u001b[0m in \u001b[0;36meigenvals\u001b[0;34m(self, error_when_incomplete, **flags)\u001b[0m\n\u001b[1;32m   1127\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1128\u001b[0m         \u001b[0mflags\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpop\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'simplify'\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m)\u001b[0m  \u001b[0;31m# pop unsupported flag\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1129\u001b[0;31m         \u001b[0meigs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mroots\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mmat\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcharpoly\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mDummy\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'x'\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0mflags\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1130\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1131\u001b[0m         \u001b[0;31m# make sure the algebraic multiplicty sums to the\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/matrices/matrices.py\u001b[0m in \u001b[0;36mcharpoly\u001b[0;34m(self, x, simplify)\u001b[0m\n\u001b[1;32m    348\u001b[0m         \u001b[0mberk_vector\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_berkowitz_vector\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    349\u001b[0m         \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_uniquely_named_symbol\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mberk_vector\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 350\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mPurePoly\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0msimplify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mberk_vector\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    351\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    352\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mcofactor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m\"berkowitz\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/matrices/matrices.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m    348\u001b[0m         \u001b[0mberk_vector\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_eval_berkowitz_vector\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    349\u001b[0m         \u001b[0mx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_uniquely_named_symbol\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mberk_vector\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 350\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mPurePoly\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0msimplify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mberk_vector\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    351\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    352\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0mcofactor\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mj\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmethod\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m\"berkowitz\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/simplify/simplify.py\u001b[0m in \u001b[0;36msimplify\u001b[0;34m(expr, ratio, measure, rational, inverse)\u001b[0m\n\u001b[1;32m    564\u001b[0m     \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mMul\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mpowsimp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    565\u001b[0m     \u001b[0m_e\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 566\u001b[0;31m     \u001b[0mexpr1\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mshorter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0m_e\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0m_mexpand\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0m_e\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m  \u001b[0;31m# issue 6829\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    567\u001b[0m     \u001b[0mexpr2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mshorter\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mtogether\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mtogether\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr1\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mdeep\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    568\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/expr.py\u001b[0m in \u001b[0;36mcancel\u001b[0;34m(self, *gens, **args)\u001b[0m\n\u001b[1;32m   3243\u001b[0m         \u001b[0;34m\"\"\"See the cancel function in sympy.polys\"\"\"\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3244\u001b[0m         \u001b[0;32mfrom\u001b[0m \u001b[0msympy\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mpolys\u001b[0m \u001b[0;32mimport\u001b[0m \u001b[0mcancel\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 3245\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mcancel\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   3246\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   3247\u001b[0m     \u001b[0;32mdef\u001b[0m \u001b[0minvert\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0mgens\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m**\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/polys/polytools.py\u001b[0m in \u001b[0;36mcancel\u001b[0;34m(f, *gens, **args)\u001b[0m\n\u001b[1;32m   6581\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Number\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mRelational\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mor\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0misinstance\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mExpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   6582\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 6583\u001b[0;31m         \u001b[0mf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfactor_terms\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;32mTrue\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   6584\u001b[0m         \u001b[0mp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mq\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mf\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_numer_denom\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   6585\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/exprtools.py\u001b[0m in \u001b[0;36mfactor_terms\u001b[0;34m(expr, radical, clear, fraction, sign)\u001b[0m\n\u001b[1;32m   1199\u001b[0m         \u001b[0;32mreturn\u001b[0m \u001b[0mrv\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1200\u001b[0m     \u001b[0mexpr\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msympify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1201\u001b[0;31m     \u001b[0;32mreturn\u001b[0m \u001b[0mdo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1202\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1203\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/exprtools.py\u001b[0m in \u001b[0;36mdo\u001b[0;34m(expr)\u001b[0m\n\u001b[1;32m   1173\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0mfactor_sum\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mexpr\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfraction\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mfraction\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msign\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0msign\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1174\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1175\u001b[0;31m         \u001b[0mcont\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mexpr\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1176\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mp\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1177\u001b[0m             \u001b[0mlist_args\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0mdo\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mAdd\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mmake_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/add.py\u001b[0m in \u001b[0;36mas_content_primitive\u001b[0;34m(self, radical, clear)\u001b[0m\n\u001b[1;32m    977\u001b[0m         \"\"\"\n\u001b[1;32m    978\u001b[0m         con, prim = self.func(*[_keep_coeff(*a.as_content_primitive(\n\u001b[0;32m--> 979\u001b[0;31m             radical=radical, clear=clear)) for a in self.args]).primitive()\n\u001b[0m\u001b[1;32m    980\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mclear\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mcon\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Integer\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mprim\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    981\u001b[0m             \u001b[0mcon\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0md\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcon\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_numer_denom\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/add.py\u001b[0m in \u001b[0;36m<listcomp>\u001b[0;34m(.0)\u001b[0m\n\u001b[1;32m    977\u001b[0m         \"\"\"\n\u001b[1;32m    978\u001b[0m         con, prim = self.func(*[_keep_coeff(*a.as_content_primitive(\n\u001b[0;32m--> 979\u001b[0;31m             radical=radical, clear=clear)) for a in self.args]).primitive()\n\u001b[0m\u001b[1;32m    980\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mclear\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mcon\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Integer\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mprim\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    981\u001b[0m             \u001b[0mcon\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0md\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcon\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_numer_denom\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/mul.py\u001b[0m in \u001b[0;36mas_content_primitive\u001b[0;34m(self, radical, clear)\u001b[0m\n\u001b[1;32m   1690\u001b[0m         \u001b[0margs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1691\u001b[0m         \u001b[0;32mfor\u001b[0m \u001b[0mi\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0menumerate\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1692\u001b[0;31m             \u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mp\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1693\u001b[0m             \u001b[0mcoef\u001b[0m \u001b[0;34m*=\u001b[0m \u001b[0mc\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1694\u001b[0m             \u001b[0;32mif\u001b[0m \u001b[0mp\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mOne\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/power.py\u001b[0m in \u001b[0;36mas_content_primitive\u001b[0;34m(self, radical, clear)\u001b[0m\n\u001b[1;32m   1598\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1599\u001b[0m         \u001b[0mb\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0me\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_base_exp\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1600\u001b[0;31m         \u001b[0mb\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0m_keep_coeff\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m   1601\u001b[0m         \u001b[0mce\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mpe\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0me\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_content_primitive\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mradical\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mclear\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0mclear\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m   1602\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Rational\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/add.py\u001b[0m in \u001b[0;36mas_content_primitive\u001b[0;34m(self, radical, clear)\u001b[0m\n\u001b[1;32m    977\u001b[0m         \"\"\"\n\u001b[1;32m    978\u001b[0m         con, prim = self.func(*[_keep_coeff(*a.as_content_primitive(\n\u001b[0;32m--> 979\u001b[0;31m             radical=radical, clear=clear)) for a in self.args]).primitive()\n\u001b[0m\u001b[1;32m    980\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mclear\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mcon\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Integer\u001b[0m \u001b[0;32mand\u001b[0m \u001b[0mprim\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    981\u001b[0m             \u001b[0mcon\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0md\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcon\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mas_numer_denom\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/operations.py\u001b[0m in \u001b[0;36m__new__\u001b[0;34m(cls, *args, **options)\u001b[0m\n\u001b[1;32m     43\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     44\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 45\u001b[0;31m         \u001b[0mc_part\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnc_part\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0morder_symbols\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mflatten\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     46\u001b[0m         \u001b[0mis_commutative\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;32mnot\u001b[0m \u001b[0mnc_part\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     47\u001b[0m         \u001b[0mobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_from_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc_part\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mnc_part\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mis_commutative\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/add.py\u001b[0m in \u001b[0;36mflatten\u001b[0;34m(cls, seq)\u001b[0m\n\u001b[1;32m    214\u001b[0m                     \u001b[0;31m# Mul, already keeps its arguments in perfect order.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    215\u001b[0m                     \u001b[0;31m# so we can simply put c in slot0 and go the fast way.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 216\u001b[0;31m                     \u001b[0mcs\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_new_rawargs\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mc\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    217\u001b[0m                     \u001b[0mnewseq\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mappend\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    218\u001b[0m                 \u001b[0;32melif\u001b[0m \u001b[0ms\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_Add\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/operations.py\u001b[0m in \u001b[0;36m_new_rawargs\u001b[0;34m(self, *args, **kwargs)\u001b[0m\n\u001b[1;32m    112\u001b[0m         \u001b[0;32melse\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    113\u001b[0m             \u001b[0mis_commutative\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_commutative\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 114\u001b[0;31m         \u001b[0;32mreturn\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_from_args\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mis_commutative\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m    115\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m    116\u001b[0m     \u001b[0;34m@\u001b[0m\u001b[0mclassmethod\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;32m~/anaconda3/lib/python3.6/site-packages/sympy/core/operations.py\u001b[0m in \u001b[0;36m_from_args\u001b[0;34m(cls, args, is_commutative)\u001b[0m\n\u001b[1;32m     60\u001b[0m             \u001b[0;32mreturn\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     61\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 62\u001b[0;31m         \u001b[0mobj\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuper\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mAssocOp\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mcls\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m__new__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcls\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0;34m*\u001b[0m\u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m     63\u001b[0m         \u001b[0;32mif\u001b[0m \u001b[0mis_commutative\u001b[0m \u001b[0;32mis\u001b[0m \u001b[0;32mNone\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m     64\u001b[0m             \u001b[0mis_commutative\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfuzzy_and\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mis_commutative\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0ma\u001b[0m \u001b[0;32min\u001b[0m \u001b[0margs\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
      "\u001b[0;31mKeyboardInterrupt\u001b[0m: "
     ]
    }
   ],
   "source": [
    "eigenvals = simplified_F_jac.eigenvals()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "e1, e2, e3 = eigenvals.keys()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "e1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sym.factor(e1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "First eigenvalue can be written as a hyperbola in $U_{gA}^*,U_{GA}^*$ plane.\n",
    "\n",
    "$$ e_1 = \\left(\\frac{\\big((R + P) - (T + S)\\big)U_{gA}^{*2} + \\big((T + S) - 2P\\big)U_{gA}^*}{N^*}\\right) - \\left(\\frac{\\big((R + P) - (T + S)\\big)U_{GA}^{*2} + \\big((T + S) - 2P\\big)U_{GA}^*}{N^*}\\right) $$\n",
    "\n",
    "### Completing the square\n",
    "\n",
    "To write this hyperbola in standard form we need to [complete the square](https://en.wikipedia.org/wiki/Completing_the_square). Completing the square for the quadratic polynomial in $U_{GA}(x^*)$ yields the following.\n",
    "\n",
    "\\begin{align}\n",
    "    \\big((R + P) - (T + S)\\big)\\left(U_{GA}^* - \\bar{U}_{GA}^*\\right)^2 - \\frac{1}{2}\\big(\\bar{N}^* - 2P\\big) \\\\\n",
    "\\end{align}\n",
    "\n",
    "where $\\bar{U}_{GA}^*$ is the value of $U_{GA}^*$ that maximizes total offspring $N$ derived above. Completing the square for the quadratic polynomial in $U_{gA}(x^*)$ yields a similar expression.\n",
    "\n",
    "\\begin{align}\n",
    "    \\big((R + P) - (T + S)\\big)\\left(U_{gA}^* - \\bar{U}_{GA}^*\\right)^2 - \\frac{1}{2}\\big(\\bar{N}^* - 2P\\big) \\\\\n",
    "\\end{align}\n",
    "\n",
    "Substituting these results into the expression for the eigenvalue yields the following.\n",
    "\n",
    "$$ e_1 = \\left(\\frac{\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big)^{2}}{\\frac{N^*}{\\big((R + P) - (T + S)\\big)}}\\right) - \\left(\\frac{\\big(U_{GA}^* - \\bar{U}_{GA}^*\\big)^{2}}{\\frac{N^*}{\\big((R + P) - (T + S)\\big)}}\\right) $$\n",
    "\n",
    "\n",
    "### Conditions for negative eigenvalue\n",
    "\n",
    "Non-invadability requires that this eigenvalue is strictly negative.\n",
    "\n",
    "$$ \\left(\\frac{\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big)^{2}}{\\frac{N^*}{\\big((R + P) - (T + S)\\big)}}\\right) - \\left(\\frac{\\big(U_{GA}^* - \\bar{U}_{GA}^*\\big)^{2}}{\\frac{N^*}{\\big((R + P) - (T + S)\\big)}}\\right) < 0$$\n",
    "\n",
    "#### Case: (R + P) < (T + S)\n",
    "In this case the inequality simplifies to the following.\n",
    "\n",
    "$$ \\big(U_{gA}^* - \\bar{U}_{GA}^*\\big)^{2} - \\big(U_{GA}^* - \\bar{U}_{GA}^*\\big)^{2} > 0 $$\n",
    "\n",
    "This implies that we have four sub-cases to consider.\n",
    "\n",
    "\\begin{align}\n",
    "    \\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) >& \\big(U_{GA}^* - \\bar{U}_{GA}^*\\big) \\\\\n",
    "    \\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) >& -\\big(U_{GA}^* - \\bar{U}_{GA}^*\\big) \\\\\n",
    "    -\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) >& \\big(U_{GA}^* - \\bar{U}_{GA}^*\\big) \\\\\n",
    "    -\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) >& -\\big(U_{GA}^* - \\bar{U}_{GA}^*\\big)\n",
    "\\end{align}\n",
    "\n",
    "These four sub-cases define the following two regions of the $U_{gA}^*,U_{GA}^*$ plane where the eigenvalue is negative.\n",
    "\n",
    "\\begin{align}\n",
    "   & 2\\big(\\bar{U}_{GA}^* - U_{gA}^*\\big) < U_{GA}^* - U_{gA}^* < 0\\\\\n",
    "   & 0 < U_{GA}^* - U_{gA}^* < 2\\big(\\bar{U}_{GA}^* - U_{gA}\\big)\n",
    "\\end{align}\n",
    "\n",
    "#### Case: (R + P) > (T + S)\n",
    "In this case the inequality simplifies to the following.\n",
    "\n",
    "$$ \\big(U_{gA}^* - \\bar{U}_{GA}^*\\big)^{2} - \\big(U_{GA}^* - \\bar{U}_{GA}^*\\big)^{2} < 0 $$\n",
    "\n",
    "This implies that we again have four sub-cases to consider.\n",
    "\n",
    "\\begin{align}\n",
    "    \\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) <& \\big(U_{GA}^* - \\bar{U}_{GA}^*\\big) \\\\\n",
    "    \\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) <& -\\big(U_{GA}^* - \\bar{U}_{GA}^*\\big) \\\\\n",
    "    -\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) <& \\big(U_{GA}^* - \\bar{U}_{GA}^*\\big) \\\\\n",
    "    -\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) <& -\\big(U_{GA}^* - \\bar{U}_{GA}^*\\big)\n",
    "\\end{align}\n",
    "\n",
    "These four sub-cases define the following two regions of the $U_{gA}^*,U_{GA}^*$ plane where the eigenvalue is negative.\n",
    "\n",
    "\\begin{align}\n",
    "   & 2\\big(\\bar{U}_{GA}^* - U_{gA}^*\\big) < U_{GA}^* - U_{gA}^* < 0\\\\\n",
    "   & 0 < U_{GA}^* - U_{gA}^* < 2\\big(\\bar{U}_{GA}^* - U_{gA}\\big)\n",
    "\\end{align}\n",
    "\n",
    "\n",
    "### Asymptotes \n",
    "\n",
    "Asymptotes of this hyperbola can be defined as follows.\n",
    "\n",
    "$$ U_{GA}^* - \\bar{U}_{GA}^* = \\pm\\big(U_{gA}^* - \\bar{U}_{GA}^*\\big) $$\n",
    "\n",
    "The above can be written as linear equations in $U_{gA}^*,U_{GA}^*$ plane.\n",
    "\n",
    "\\begin{align}\n",
    "    U_{GA}^* =& U_{gA}^* \\\\\n",
    "    U_{GA}^* =& 2\\bar{U}_{GA}^* - U_{gA}^*\n",
    "\\end{align}\n",
    "\n",
    "Note that the asymptotes of this hyperbola correspond to the locus of points for which the value of the first eigenvalue is exactly zero.\n",
    "\n",
    "#### Discussion\n",
    "\n",
    "Suppose $U_{GA}^* = \\bar{U}_{GA}^*$, then both of the sets of inequalities will hold for any value of $U_{gA}^*$. This implies that the eigenvalue will be negative for any value of $U_{gA}^*$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "# vectorized numeric repr for the eigenvalue\n",
    "_numeric_e1 = sym.lambdify((UGA_star, UGA_prime_star, T, R, P, S, M), e1, modules=\"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "def plot_first_eigenvalue(T, R, P, S, M):\n",
    "\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10, 10))\n",
    "    ax.set_ylabel(r\"$U_{GA}^*$\", fontsize=20, rotation=\"horizontal\")\n",
    "    ax.set_xlabel(r\"$U_{GA}^{'*}$\", fontsize=20)\n",
    "    ax.set_title(r\"Eigenvalue, $e_1$\", fontsize=25)\n",
    "    ax.grid(\"off\")\n",
    "\n",
    "    equilibrium_selection_probs = np.linspace(0, 1, 100).reshape(-1, 1)\n",
    "    equilibrium_selection_derivs = np.linspace(0, 10, 1000).reshape(1, -1) # sensible UGA'(0) >= 1\n",
    "    Z = _numeric_e1(equilibrium_selection_probs, equilibrium_selection_derivs, T, R, P, S, M)\n",
    "    cax = ax.imshow(Z, origin=\"lower\", aspect=\"auto\", vmin=-0.5, vmax=1.0)\n",
    "\n",
    "    levels = np.arange(-0.25, 1.25, 0.25)\n",
    "    contours = ax.contour(Z, levels=levels, colors='w', origin='lower')\n",
    "    ax.clabel(contours, contours.levels, inline=True, fontsize=10)\n",
    "    \n",
    "    # adjust the tick labels\n",
    "    locs, _ = plt.xticks()\n",
    "    plt.xticks(locs[1:], np.linspace(0, 10, locs.size-1))\n",
    "    locs, _ = plt.yticks()\n",
    "    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mpld3.disable_notebook()  # don't need interactive plotting for this!\n",
    "\n",
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "M_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$M$\")\n",
    "\n",
    "w = widgets.interactive(plot_first_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, M=M_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "UGA_prime_star, = sym.solve(numerator, UGA_prime_star)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sym.factor(UGA_prime_star)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "After a bit of fiddling, we arrive at the same stability condition derived above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "   % U'_{GA}(x^*) < -\\frac{(PU_{GA}(x^*)^2 - 2PU_{GA}(x^*) + P + RU_{GA}(x^*)^2 - SU_{GA}(x^*)^2 + SU_{GA}(x^*) - TU_{GA}(x^*)^2 + TU_{GA}(x^*))^2}{(2PRU_{GA}(x^*)^2 - 2PRU_{GA}(x^*) - PSU_{GA}(x^*)^2 + 2PSU_{GA}(x^*) - PS - RTU_{GA}(x^*)^2)} \\\\\n",
    "    U'_{GA}(x^*) < \\frac{\\bigg(\\big((R + P) - (T + S)\\big)U_{GA}(x^*)^2 + \\big((T + S) - 2P\\big)U_{GA}(x^*) + P\\bigg)^2}{\\big(R(T-2P) + PS\\big)U_{GA}(x^*)^2 + 2P(R - S)U_{GA}(x^*) + PS}\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "e2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$ e_2 = \\left(\\frac{\\big((R + P) - (T + S)\\big)U_{gA}^{*2} + \\big((T + S) - 2P\\big)U_{gA}^* + P}{4N^*}\\right) - \\frac{3}{4} $$\n",
    "\n",
    "### Completing the square\n",
    "\n",
    "Completing the square for the quadratic polynomial in $U_{gA}(x^*)$ yields the following expression.\n",
    "\n",
    "\\begin{align}\n",
    "    \\big((R + P) - (T + S)\\big)\\left(U_{gA}^* - \\bar{U}_{GA}^*\\right)^2 + \\frac{1}{2}\\bar{N}^* \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally we need to find conditions under which the third eigenvalue is strictly negative."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'UgA_star' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "\u001b[0;32m<ipython-input-14-e310acacc8ef>\u001b[0m in \u001b[0;36m<module>\u001b[0;34m()\u001b[0m\n\u001b[1;32m      1\u001b[0m \u001b[0;31m# vectorized numeric repr for the eigenvalue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0m_numeric_e2\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msym\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mlambdify\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mUGA_star\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mUgA_star\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mT\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mR\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mP\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mS\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mM\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mm\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0me2\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmodules\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m\"numpy\"\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m: name 'UgA_star' is not defined"
     ]
    }
   ],
   "source": [
    "# vectorized numeric repr for the eigenvalue\n",
    "_numeric_e2 = sym.lambdify((UGA_star, UgA_star, T, R, P, S, M, m), e2, modules=\"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "def plot_second_eigenvalue(T, R, P, S, M, m):\n",
    "\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10, 10))\n",
    "    ax.set_ylabel(r\"$U_{GA}^*$\", fontsize=20, rotation=\"horizontal\")\n",
    "    ax.set_xlabel(r\"$U_{gA}^{*}$\", fontsize=20)\n",
    "    ax.set_title(r\"Eigenvalue, $e_2$\", fontsize=25)\n",
    "    ax.grid(\"off\")\n",
    "\n",
    "    equilibrium_selection_probs = np.linspace(0, 1, 100)\n",
    "    UgAs, UGAs = np.meshgrid(equilibrium_selection_probs, equilibrium_selection_probs)\n",
    "    Z = _numeric_e2(UGAs, UgAs, T, R, P, S, M, m)\n",
    "    cax = ax.imshow(Z, origin=\"lower\")\n",
    "    contours = ax.contour(Z, colors='w', origin='lower')\n",
    "    ax.clabel(contours, contours.levels, inline=True, fontsize=10)\n",
    "    \n",
    "    # adjust the tick labels\n",
    "    locs, _ = plt.xticks()\n",
    "    plt.xticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "    locs, _ = plt.yticks()\n",
    "    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mpld3.disable_notebook()  # don't need interactive plotting for this!\n",
    "\n",
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "M_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$M$\")\n",
    "m_slider = widgets.FloatSlider(value=0, min=0, max=100, step=0.1, description=r\"$m$\")\n",
    "\n",
    "w = widgets.interactive(plot_second_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, M=M_slider, m=m_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "e3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "# vectorized numeric repr for the eigenvalue\n",
    "_numeric_e3 = sym.lambdify((UGA_star, UgA_star, T, R, P, S, M, m), e3, modules=\"numpy\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "def plot_third_eigenvalue(T, R, P, S, M, m):\n",
    "\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10, 10))\n",
    "    ax.set_ylabel(r\"$U_{GA}^*$\", fontsize=20, rotation=\"horizontal\")\n",
    "    ax.set_xlabel(r\"$U_{gA}^{*}$\", fontsize=20)\n",
    "    ax.set_title(r\"Eigenvalue, $e_3$\", fontsize=25)\n",
    "    ax.grid(\"off\")\n",
    "\n",
    "    equilibrium_selection_probs = np.linspace(0, 1, 100)\n",
    "    UgAs, UGAs = np.meshgrid(equilibrium_selection_probs, equilibrium_selection_probs)\n",
    "    Z = _numeric_e3(UGAs, UgAs, T, R, P, S, M, m)\n",
    "    cax = ax.imshow(Z, origin=\"lower\")\n",
    "    contours = ax.contour(Z, colors='w', origin='lower')\n",
    "    ax.clabel(contours, contours.levels, inline=True, fontsize=10)\n",
    "    \n",
    "    # adjust the tick labels\n",
    "    locs, _ = plt.xticks()\n",
    "    plt.xticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "    locs, _ = plt.yticks()\n",
    "    plt.yticks(locs[1:], np.linspace(0, 1, locs.size-1))\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "mpld3.disable_notebook()  # don't need interactive plotting for this!\n",
    "\n",
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=25, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=3, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=2, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "M_slider = widgets.FloatSlider(value=1, min=0, max=100, step=0.1, description=r\"$M$\")\n",
    "m_slider = widgets.FloatSlider(value=0, min=0, max=100, step=0.1, description=r\"$m$\")\n",
    "\n",
    "w = widgets.interactive(plot_third_eigenvalue, T=T_slider, R=R_slider, P=P_slider, S=S_slider, M=M_slider, m=m_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## When is the fitness-maximizing interior equilibrium un-invadable by randomista?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "evaluated_e3 = e3.subs({UGA_star: optimal_UGA_star, UgA_star: optimal_x1_star, m: 0})\n",
    "simplified_e3 = sym.factor(sym.cancel(sym.together(evaluated_e3)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "numerator, denominator = sym.fraction(simplified_e3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "numerator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\begin{align}\n",
    "    4\\bigg(-\\big(P^3 + 3P^2R + 3PR^2 + R^3\\big) - \\big(P^3 - 3P^2S + 3PS^2 + S^3\\big) + \\big(-P^3 + 3P^2T - 3PT^2 + T^3\\big) - \\big(R^3 - 3R^2S + 3RS^2 - S^3\\big) + \\big(T^3 - 3RT^2 + 3R^2T - R^3\\big) + 2P^3 + 2R^3 + 8PRS + 8PRT - 8PST - 8RST - 2T^3 + \\big(S^3 + 3S^2T + 3ST^2 + T^3\\big)\\bigg)M^3 + 2\\big(4P^3R + 8P^2R^2 - 8P^2RS - 8P^2RT - P^2S^2 - 2P^2ST - P^2T^2 + 4PR^3 - 8PR^2S - 8PR^2T + 2PRS^2 + 4PRST + 2PRT^2 + 2PS^3 + 6PS^2T + 6PST^2 + 2PT^3 - R^2S^2 - 2R^2ST - R^2T^2 + 2RS^3 + 6RS^2T + 6RST^2 + 2RT^3 - S^4 - 4S^3T - 6S^2T^2 - 4ST^3 - T^4\\big)M^2 + \\big(- 4P^3R^2 - 4P^2R^3 + 4P^2R^2S + 4P^2R^2T + 2P^2RS^2 + 4P^2RST + 2P^2RT^2 + 2PR^2S^2 + 4PR^2ST + 2PR^2T^2 - 2PRS^3 - 6PRS^2T - 6PRST^2 - 2PRT^3 - 0.25PS^4 -PS^3T - 1.5PS^2T^2 - PST^3 - 0.25PT^4 - 0.25RS^4 - RS^3T - 1.5RS^2T^2 - RST^3 - 0.25RT^4 + 0.25S^5 + 1.25S^4T + 2.5S^3T^2 + 2.5S^2T^3 + 1.25ST^4 + 0.25T^5\\big)M + \\big(- P^2R^2S^2 + 2P^2R^2ST - P^2R^2T^2 + P^2RS^3 - P^2RS^2T - P^2RST^2 + P^2RT^3 - 0.25P^2S^4 + 0.5P^2S^2T^2 - 0.25P^2T^4 + PR^2S^3 - PR^2S^2T - PR^2ST^2 + PR^2T^3 - PRS^4 + 2PRS^2T^2 - PRT^4 + 0.25PS^5 + 0.25PS^4T - 0.5PS^3T^2 - 0.5PS^2T^3 + 0.25PST^4 + 0.25PT^5 - 0.25R^2S^4 + 0.5R^2S^2T^2 - 0.25R^2T^4 + 0.25RS^5 + 0.25RS^4T - 0.5RS^3T^2 - 0.5RS^2T^3 + 0.25RST^4 + 0.25RT^5 - 0.0625S^6 - 0.125S^5T + 0.0625S^4T^2 + 0.25S^3T^3 + 0.0625S^2T^4 - 0.125ST^5 - 0.0625T^6\\big) \\\\\n",
    "    4\\bigg(-\\big(R + P\\big)^3 - \\big(P - S\\big)^3 + \\big(T - P\\big)^3 - \\big(R - S\\big)^3 + \\big(T - R\\big)^3 + 2P^3 + 2R^3 + 8PRS + 8PRT - 8PST - 8RST - 2T^3 + \\big(T + S\\big)^3\\bigg)M^3 + 2\\big(4P^3R + 8P^2R^2 - 8P^2RS - 8P^2RT - P^2S^2 - 2P^2ST - P^2T^2 + 4PR^3 - 8PR^2S - 8PR^2T + 2PRS^2 + 4PRST + 2PRT^2 + 2PS^3 + 6PS^2T + 6PST^2 + 2PT^3 - R^2S^2 - 2R^2ST - R^2T^2 + 2RS^3 + 6RS^2T + 6RST^2 + 2RT^3 - S^4 - 4S^3T - 6S^2T^2 - 4ST^3 - T^4\\big)M^2 + \\big(- 4P^3R^2 - 4P^2R^3 + 4P^2R^2S + 4P^2R^2T + 2P^2RS^2 + 4P^2RST + 2P^2RT^2 + 2PR^2S^2 + 4PR^2ST + 2PR^2T^2 - 2PRS^3 - 6PRS^2T - 6PRST^2 - 2PRT^3 - 0.25PS^4 -PS^3T - 1.5PS^2T^2 - PST^3 - 0.25PT^4 - 0.25RS^4 - RS^3T - 1.5RS^2T^2 - RST^3 - 0.25RT^4 + 0.25S^5 + 1.25S^4T + 2.5S^3T^2 + 2.5S^2T^3 + 1.25ST^4 + 0.25T^5\\big)M + \\big(- P^2R^2S^2 + 2P^2R^2ST - P^2R^2T^2 + P^2RS^3 - P^2RS^2T - P^2RST^2 + P^2RT^3 - 0.25P^2S^4 + 0.5P^2S^2T^2 - 0.25P^2T^4 + PR^2S^3 - PR^2S^2T - PR^2ST^2 + PR^2T^3 - PRS^4 + 2PRS^2T^2 - PRT^4 + 0.25PS^5 + 0.25PS^4T - 0.5PS^3T^2 - 0.5PS^2T^3 + 0.25PST^4 + 0.25PT^5 - 0.25R^2S^4 + 0.5R^2S^2T^2 - 0.25R^2T^4 + 0.25RS^5 + 0.25RS^4T - 0.5RS^3T^2 - 0.5RS^2T^3 + 0.25RST^4 + 0.25RT^5 - 0.0625S^6 - 0.125S^5T + 0.0625S^4T^2 + 0.25S^3T^3 + 0.0625S^2T^4 - 0.125ST^5 - 0.0625T^6\\big) \\\\\n",
    "\\end{align}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "(S - R)(S^2 - 2SR + R^2) = S^3 - 3RS^2 + 3SR^2 - R^3"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sym.factor(-P**3 - 3*P**2*R + 3*P**2*S + 3*P**2*T - 3*P*R**2 + 8*P*R*S + 8*P*R*T - 3*P*S**2 - 8*P*S*T - 3*P*T**2 - R**3 + 3*R**2*S + 3*R**2*T - 3*R*S**2 - 8*R*S*T - 3*R*T**2 + S**3 + 3*S**2*T + 3*S*T**2 + T**3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "r1, r2, r3 = sym.solve(numerator, M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "sym.cancel(sym.together(r1))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "denominator"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Denominator will be negative if and only if...\n",
    "\n",
    "\\begin{align}\n",
    "    % 4MP + 4MR - 4MS - 4MT - 4PR + S^2 + 2ST + T^2 < 0 \\\\\n",
    "    % 4M\\big((R + P) - (T + S)\\big) - 4PR + (T + S)^2< 0 \\\\\n",
    "    % 4M\\big((R + P) - (T + S)\\big) < 4RP - (T + S)^2 \\\\\n",
    "    M < \\frac{1}{4}\\frac{4RP - (T + S)^2}{(R + P) - (T + S)} \\\\\n",
    "    M < \\frac{RP - \\left(\\frac{T + S}{2}\\right)^2}{(R + P) - (T + S)} \\\\\n",
    "\\end{align}\n",
    "\n",
    "...since we are looking at interior equilibrium the RHS of the above inequality will be positive."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": [
    "_f = sym.lambdify((T, R, P, S, M), evaluated_e3, modules = \"numpy\")\n",
    "\n",
    "def plot_non_invadable_by_randomista(T, R, P, S):\n",
    "    fig, ax = plt.subplots(1, 1, figsize=(10,6))\n",
    "    Ms = np.linspace(0, min(0.5 * (T + S), P), 100)\n",
    "    ax.plot(Ms, _f(T, R, P, S, Ms))\n",
    "    ax.set_xlabel(r\"$M$\", fontsize=15)\n",
    "    ax.set_ylabel(r\"$e_3$\", rotation=\"horizontal\", fontsize=15)\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# sliders used to control the Prisoner's Dilemma Payoffs\n",
    "T_slider = widgets.FloatSlider(value=10, min=0, max=100, step=0.1, description=r\"$T$\")\n",
    "R_slider = widgets.FloatSlider(value=8, min=0, max=100, step=0.1, description=r\"$R$\")\n",
    "P_slider = widgets.FloatSlider(value=6, min=0, max=100, step=0.1, description=r\"$P$\")\n",
    "S_slider = widgets.FloatSlider(value=4, min=0, max=100, step=0.1, description=r\"$S$\")\n",
    "\n",
    "w = widgets.interactive(plot_non_invadable_by_randomista, T=T_slider, R=R_slider, P=P_slider, S=S_slider)\n",
    "display(w)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "solutions = sym.solve(simplified_e3, M)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true,
    "jupyter": {
     "outputs_hidden": true
    }
   },
   "outputs": [],
   "source": []
  }
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