Vectors.ipynb

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    "%matplotlib inline\n",
    "#%matplotlib notebook\n",
    "\n",
    "import matplotlib\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.ticker as ticker\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "import matplotlib.patches as mpatches\n",
    "from matplotlib.patches import Arc\n",
    "from mpl_toolkits import mplot3d\n",
    "from matplotlib.pyplot import *\n",
    "from matplotlib import collections as mc"
   ]
  },
  {
   "cell_type": "markdown",
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   "source": [
    "# Vectors"
   ]
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   "source": [
    "## Introduction\n",
    "\n",
    "Hi there! Welcome to the guide on vectors!\n",
    "\n",
    "Vectors are a very important concept in linear algebra and serve as a fundamental primitive. They are used in various ways throughout linear algebra, calculus, physics, and many other domains. As a result, introducing vectors serves as a logical starting point in our exploration of linear algebra.\n",
    "\n",
    "What are vectors? **Vectors are simply a list of numbers that have some meaning**. For example, in physics, velocity can be represented as a vector. In chemistry, the number of molecules composing a compound can be expressed as a vector. In sociology, the results of a survey can be expressed as a vector. Let's see an example of a vector consisting of 2 elements:\n",
    "\n",
    "$$\\begin{bmatrix}5 \\\\ 2\\end{bmatrix}$$\n",
    "\n",
    "Since a vector is just a list of numbers, it can contain any number of elements. More generally, we can express a vector with $i$ numbers as:\n",
    "\n",
    "$$\\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\vdots \\\\ v_i\\end{bmatrix}$$\n",
    "\n",
    "Where $v_1, v_2, \\dots v_i$ are real numbers. Vectors (and matrices as we'll see later) are represented as a list of numbers surrounded by square brackets, and each number in the vector is commonly called a **component** or **element**.\n",
    "\n",
    "When it comes to thinking about vectors, we can reason about them in a couple of different ways: one, we can imagine a vector as being a point in space. From this perspective, we allow the components of the vector to make up the \"axes\" of the point. Second, we can think of a vector as something with a length and direction. Think of it as an arrow of sorts with a certain length and pointing off in some direction. Don't worry if this doesn't make sense—both will be covered below.\n",
    "\n",
    "## Vectors as a point in space\n",
    "\n",
    "A common way to reason about a vector is by viewing it as a point in space. In this way, we allow the components of the vector to make up the dimensions of the point. For example, given the the following vector:\n",
    "\n",
    "$$\\begin{bmatrix}2 \\\\ 1\\end{bmatrix}$$\n",
    "\n",
    "We can say this is a 2-dimensional vector, with $x = 2$ and $y = 1$. Similarly, we can have a 3-dimensional vector:\n",
    "\n",
    "$$\\begin{bmatrix}4 \\\\ 1 \\\\\\ -2\\end{bmatrix}$$\n",
    "\n",
    "With $x = 4$, $y = 1$, and $z = -1$.\n",
    "\n",
    "A benefit of this thinking is that it allows us to visualize vector by viewing them geometrically and plotting them in a coordinate system. For example, let's take the vector from above and visualize it:\n",
    "\n",
    "$$\\begin{bmatrix}2 \\\\ 1\\end{bmatrix}$$\n",
    "\n",
    "Since the vector above has 2 components, we can visualize it in the X-Y plane by allowing the first component, $2$, to be the X-coordinate and the second component, $1$, to be the Y-coordinate:"
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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(6, 6))\n",
    "axhline(y=0, color='k', linewidth=0.5)\n",
    "axvline(x=0, color='k', linewidth=0.5)\n",
    "grid(linestyle='dotted')\n",
    "axis([-1, 3, -1, 3])\n",
    "annotate('Vector [2, 1] representing x = 2, y = 1', xy=(0.5, 1.2), ha='left')\n",
    "quiver([0], [0], 2, 1, angles='xy', scale_units='xy', scale=1)\n",
    "plot([0, 2, 2, 2], [0, 0, 0, 1], 'b--')\n",
    "xticks([-1, 0, 1, 2, 3])\n",
    "yticks([-1, 0, 1, 2, 3])\n",
    "annotate('x = 2', xy=(1, -0.25), ha='center', color='blue')\n",
    "annotate('y = 1', xy=(2.1, 0.5), ha='left', color='blue')\n",
    "show()"
   ]
  },
  {
   "attachments": {
    "image.png": {
     "image/png": 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p1a8eGJaNCCCAAAIIILCAAiQOFhCbXSGAAAIILJyAT3T49cceS1fed19SokBJg6fs8x7767cm+Ru2LNWJDof1u+3t8ZYGnxdB672WPBi2XHLjjdPzI9hhqJIDJ500mRg48A0PV095OObGY9K3v53SPfdMvnTrAomDYbJsRwABBBBAYOEESBwsnDV7QgABBBBYQAGf6PB8+9O2XoMmOix1aylPdFjq83yX15MIl199ddIVCaVlm11K4Lc12N0N6YQTUjr88JQ2v+K2tPKxVem9p56a0k9MJgxsXsX0gGV8dBsDCwIIIIAAAggsHQESB0vnWNATBBBAAIF5FBg00eE87rZx6JMOOKBx3aVSMSYRSn3abokDnxfhuhM3ppcffEGyJzimjY8/np6lDMLUomSBblfQvAfLkMKHwRoBBBBAAIFOCpA46ORhZVAIIIBAPwSu2b27ug1h944d6f991rNGHvRSm+jwMs0MaMuwxyOOPNDFbGCZgL1PP5122UQGurXhytVfSiv32ASLjxyQfvHcc6ue/dVfpXTzzZMTJJ544uRVCd7l37X7F0458MC02yZA+IVjj/Vi1ggggAACCCCwgAIkDhYQm10hgAACCMxdoJro0OYr+Ko9z2+PJQyqSQ5t7oJ32qx6/3XIRH3a+3KY6FBPc+jKorH4vAgak97vsYTPbptn4s//PFVzG2hCxB/+MKXjj0/pFa9I6aKLJkf/W7fcUl2tcLslDXTVwu9u3pxOtiSCZSDSy+32kxOHPNGhK4aMAwEEEEAAgcUWIHGw2EeA/SOAAAJLQGCPzZT/tP1VeJWdnK3Us/GW6KKJDq+6//60wxIFShhst8/6S7RPdDjoL/VL7eqCJUo8L92KtzQocaBFT2nYaW/tUFYvyyUkyyUkO6zTyzZ7RqMe91i9bIKElfZSEkHH+66nnqquRPjZdeuq+jfbEzPOPOqo6ba8QQABBBBAAIH2BEgctGdJJAQQQGBZCShZoMXXer9bZ2+2aD6ApZhE0CSH/+Paa6ukQdVR+8eTBv65tGaiw5LMwpYriaClWtucinZI08knp/Td705OivjNb6b0m7+Z0vveZ0kEm/hgv9Wrq8TBypBAUCJBSYSf1X0NUwtJA5dgjQACCCCAQPsCJA7aNyUiAgggsGQFdFWB7jOPyYJcZ1WnnkRQvaVwNcJHLrywmgMg12/Klp/AG9+Y0lVXpXTwwSl95zsp3XRTSn4Hgh6bqStKVthVMEoQeRJBT2lQIqHJcrfF4JaGJlLUQQABBBBAoCxA4qBswxYEEECgMwJKFAxLFpQGW08iqN58XI1wjfVRtyHsskvQuzDRYcmzz+UTEymtXz9bwHJBSa+PfWwycWB3HaQvf3lyPopb7RmNur3habsaZsdUEkHzHVz+Uz81O1Ct5EM2cYISDqvXrEknWPLhyx9ZkT7x23ZvRFgmJnbavief/3jBBautf/vba3WowVsEEEAAAQQQIHHAdwABBBDoqIAnCnzdxjCVRNBSvxph3CsRfKLDK++7L+nJCDun5i7o0kSHbbh3IcZXvpKSzWeZPvWplI45JqXc/I+HHjr5KEY7168ey+i3l8R5EfR+79RtNoNcLrHLFzZ9+0CbVHFPetbLdqbbLXFwxttWpVdv35v+4SMzW05M2OQKtmh96aUpXXLJwWnDBrsEggUBBBBAAAEEKgESB3wREEAAgY4J7NLl3VNzFSzE0HxfSh7sP3X5eJNEwtcffzxdZQkDfypCNdGh9b1JWyY6XIgj2+4+bNqMZHNZpltvTUnTHOQSB5o3UXMeaI7DV7963/5nzItgxUoeXH711eldL7FJEgrLzVftSH/xLstQTC2nvmh3eveXDrAkwvZ0ygtPT1/5yL3pzmu2++YZ60sv1VUJj1vy4LgZ5XxAAAEEEECgrwIr7K9Hk38+6qsA40YAAQQ6IKBkgRY/iV/sIfnJvxIJ/j7Xp//brl3XVQalpUuPJSyNsU/lH/94Sv/4j9XTFNP556d02mkpveY1KV1xRUq33ZbSPfekpOTBKaek9B//49xkXvwLV6dvfU4TMR6efvL/WZUueNvjSXMmVE/ksO/cbd9I6a5r16R/+cuTMjvabLcr7LC5F8qJiUwjihBAAAEEEOisAFccdPbQMjAEEOi6gCY61G0ISyVZEL3VNy077PYDLUoe5JIIH7Eb3vUIxdzSt6TBJrP6wg9+kN5+zjkzOIb9ZX1G5SX+QQ9BOOCAlP71X1PavDklm3ogfeITk5MhKvd15JEpPfvZKT33uW0NxC5xSFvTFb+zLp33v26pklRf+/gRafNNx9nL7ouYsTxun1Rfi2UwkpIOLAgggAACCCAgARIHfA8QQACBZSagqwt0Yu4n50ux+/faxWx//cgj6WKdCdqivtaTCCovXY2gWxH6uOgS/HoixS/T74KH5jP83vdSUpLgwQcn13pvT1yskga60uBHfmTyKoS5jlffoW+lOy3MNns9kD704hXpRT+f7CoEfVaSQIuSB3rvn1XGggACCCCAAAJ1ARIHdRE+I4AAAktQYKndipAj+p49Iu9JeyLC12xKfE10uGPbtvR+u6F9xxNPpN/RlPlTSz2J4OWeLPAJ8bycdbcE3v3uyfF8/vMp2UMSqqTBQQel9KpXjT/Ou21Ojzvt9oOXH374wCCTiQNV8USBr2c3u+CCtbMLKUEAAQQQQKCnAiQOenrgGTYCCCwPgfme6LCNJy58264m+Majj854KsJT9ln3k6+wWxQGLUoivPqZz6xefvWByvz9oLZsW94Cr3/93Po//UQOe1TDbstAaK6Ml7/gBdNBN2xYZ/MUrLZHLV47XaY3z3ie3SZx04yiWR8uueQMmxjxzFnlFCCAAAIIINBXARIHfT3yjBsBBJaswChXF7Rx4j9XiOdbcuDv9XQEO3HTxHO+xKTBoH6+8uSTq7kavI7Gv8qex1eaF8Hjs+6nwNXbt6er7r8/7VGywK5q0XfOn8ih2zzi3Bjr1x+a9u5db0mAm9PGjQ9bwmBLNmmwfv26FK8wIGnQz+8Wo0YAAQQQKAuQOCjbsAUBBBBYMAH9lV0nznGiQz+RXrBOzGFHv37WWem9GzfOIcLMphq7Xp5EUSJhtW6Et0XvWfolcL0N99ypIf+9PX6hSlJZ0iA+kWPQVSpKBHzp5pvTa898Sdqw/ZF06aX2aIc0+T1S0oCnJ/Tr+8RoEUAAAQRGFxh8Deno8WiBAAIIIDCCwE77q+lTNi/ANv3l1OYF8BPmpZY0+Dvr47DlgxdckK3yKs14N8dFHnKKr6VmNMch0nyAgCcNVGWDPXLh0akrXAY0mbWJuTNmkVCAAAIIIIBAYwGuOGhMRUUEEECgHQElC7T4up2o7Ub5gf11/0lLZnx969bpuQuus8+XnHdeox15suAnTj21Uf1RKnnCQEkELX41AlcijKK4dOvebUmiE4dcVfKxN7xh1tMnlu6I6BkCCCCAAALLX4DEwfI/howAAQSWiYASBUshWeAn3jm2G+3JCN9QssDuI9dTEXQpeJzoULcj/NbLX55rmn7ja19Lmq9AL1/q+5qPk3vto55E0P7nY18+LtbtCtxo4ZSoutKuJNgz9USO3/6xH2u8E57I0ZiKiggggAACCIwlQOJgLDYaIYAAAs0EPFHg62atmtWqn5Q3azW41tn2WLv/abPUV5PO2WMUfYkTHXpZfV1KKMR6c+lzk0RAPYmgfWtuhCZtYz8X4/1JBxywGLtd1H1qosOJBx6YfiqCT3SoJ3K805IIcaLDXEeVMOAWhJwMZQgggAACCLQrQOKgXU+iIYAAApXAXK4umMvJ9Vz59+7dm/6DnYy97+tfL4ZSnSbLihUrmlRrXGcUFyUKvH79aoSlnES47JWvbOyxXCvOmuhw6sqW0hM5cuPURIfDkgq5dpQhgAACCCCAwHgCJA7Gc6MVAgggMEtAyQKdrPoJ66wKUwXDtpfalcqbnsh7+y9aP1839YQCL2u6/s3zz29a1R6D1yzB0CTgqEmInLHK/MoPXYXAUxqayLdfpz7R4Tv+7u/SyiFzGtR7wVUGdRE+I4AAAgggML8CJA7m15foCCDQcQE/EfW1Dzd34urbmq7bOvG+Zc2a9IRNJPiNLVuqpIbmLbjBXr/+vOcVu3LhSSelqzZtqrbrvfqitR4bWVoGPQ6v1KZp+agWwxINOl5+zPazeR10FYLWerEsrMCoSYOF7R17QwABBBBAAAEJ8AuJ7wECCCAwhoCfeI6bIBj1RLhJF+sn9d/Zf//0z489lnZbwsCfe7/j8cfTLruvXHMW1OvHfShpsP7EE6tkQSwf9H5QvEHtmmwbNSnRxNeTC7ttXge99DhMTxwcYPMN+Psm/aPOPgGf6PAqm6NgtxI0lqT6rQETHeqWg3d+8YvTAZjocJqCNwgggAACCCwZARIHS+ZQ0BEEEFjqAtvthFuLTjCbLE1OXgfFGedE/OYDD0xnTj2m8EdtgrkrbKJD3Tte3T8+deuAT3R4yT//c7r0pS/NdiFXPmw8fiKeDTjHwlEtmiQacuPZZWZatNZ4lDwgidDs4Gmiw40PPph22f8+NLlmfCJHGpA48KQBEx02c6YWAggggAACiyFA4mAx1NknAggsKwElDAYlC3InoE0GOOrJ8LCY6scZdsIWZxZ49zOfmTZcfXWx6bh9zwVsM1YuvsqaJiea2g5KMGg8SiB4MkFJhDV224cWrkaoGNINtjrHXh+8667Jq1osSZV7IoeSA6XJDEkYTFryLwIIIIAAAktZgMTBUj469A0BBBZNIHd1wagnxk1PXkuDHHV/pTil8g0veUlp06JObFjslG0Y1WRYomHYMYqJBd3O8MTUIyo9caBEgr8f1O+ublPSQMt7TzmlWjPRYcXAPwgggAACCHROgMRB5w4pA0IAgXEF/F53Txo0jTPs5DMXZ9QTYI9xm92KoIkO//mRR9LTui/f/sL7f9lJW9N4FzzjGVUozV/QtI3ve9z1fOxnWELA+9pk34NilY6t5rhQUiEmEvqeRJA5Ex36N481AggggAAC3RIgcdCt48loEEBgDAElCpokC0onkaVdNjlpbdr2uzZZ3zc1saElDKpLwTVvgT7b/eQbbB6DS1784lKoqlwJAyULcsuo48rFGKcs/jV/1Paj2g5KDjSJlWsf3XyyTI1DVyAcdNBB1ZC6cDXCP1ui6qubN1eJqvc/5zkjHSomOhyJi8oIIIAAAggsWQESB0v20NAxBBCYTwFPFPg67iueEMby3PsmJ531dk3b3Gonn6fbnAVaJu6+u7qHXEmD7Xa5/N49e6onI/gJbanPG+2E730vfGEVo1Sn2rgI/7TZn2FJiKbm7lnnGNS+3kZXrjxmT7PQ4omDA+1Kkf3tKReDlk2WBPpPX/96uuyVr5xR7XKbo+JdA24rmVG5pQ/X2Biu1FMRrE+6qqV6Kod97y6+5Zb00de/fuhemLdgKBEVEEAAAQQQWFYCJA6W1eGiswggMFeB+tUFTU9eB5045vo0av0Yw/t0mp2oPT214e3r1qUP3HprlSxQkT8ZIbbLvfdbE3zbXPrlMRZiXT8ZH7ZPNxtWT9sHJRmG+eT6NaiNkghaHrerQ7QokTAsieBPGaga2D9n2LFfiKU+0eH0lS2WOLD7WqouDPvekTBYiCPFPhBAAAEEEFh4gZULv0v2iAACCCysgCcLHn744bTN/mKvk0x/1Xuik8Dcq15Pn3P1vCxXX2W+30HrUtv3D7gdwfc7bF2KvdTKh41j2PZB4xlkH7flYoy633p9PaFBVyPou6h1fGpDbn8LWRYnOnz4hz9Mj9xzT9qhySCnkgbel3piw8u1fu2ZZ8aPvEcAAQQQQACBjghwxUFHDiTDQACBmQI6IdNfe5+y+7NLi07qmixN6ymWTjybLlvt0vW77ZaDq+0EUhMd6nLwi084oWnzdMqhh1avHy+0GaUvjXe6yBUHXS0QuzbKMctdRaBYw/xyfRm037gf1YtJA93GsGvVqjiERX3/X37qp1IpQVB6rOKidpidI4AAAggggMC8CpA4mFdegiOAwEILKFHgSYO470EndF6vSR3VHXZC6fHiOsa++eCD0532l9xNW7dO3j+ueQs00aFN0vhb9lfe33jBC2LT6n3c5yuOPz7p5Uvc5mVdXY871txJvhvFY+Nl9XU86fdtw/pS32dpP4qt7+w2m09gvhdNdHilfcf2WKLq/c9+9ki784kOR2pEZQQQQAABBBDohACJg04cRgaBQL8F/KoCX5dO0Fxp2HavN+zE0OtpPSzm7ZYsOE33ittylU10GCece7rhRIf/ZE9P+PXnP7+KUfpnWD9K7ZZzee6kvj6eUY5l/YRfsZq41vsxaJ9xH9OxC1fA+JUJwyZXrI/ZP2uiw6vuv79KTE3PW2CJqmqiw5/+aa9WXDNvQZGGDQgggAACCPRGgMRBbw41A0WgewJKFGjOgkHL9EnZgEqDTvBisyaxYv0Y91Q7UfObGH7l8MPTb9vs9D7RnK9jW72v7+/lxx03q6zeZpTPsX+jtFuIuvHEetj+6k7D6vv2+om+lzdxyfVvUD/q+8rt4+lC4kD9evTRR6vuKXngj3oclEi40Wqfba/PWcLgFptPQbfBlJ7IUQXO/EPCIINCEQIIIIAAAj0VIHHQ0wPPsBFYrgKeKPCrC+I4Bp24eb3cCZtv83WTOF63STyv62vFLyULVKfJ/sfZr+9/OazbGl/uBN/H38RZdesn/Spr0r+470H7ysXXPnyJbXX1QUwiqI4SCfUkgpIGWn7+2GPTv7/uurTH2vl3zteTNcr/MtFh2YYtCCCAAAII9E2AxEHfjjjjRWAZCvil2vVkQTyhyg1r2MndsPYec1gcr/cde9TeNTZ3gU7S9Bfet9tJW2l573nnpQ/aCZ0vJx9ySDrJbmfQVQVN9+dtS+u24pTiL4XyeHKe68+oBrl4Tb4nuZP/QfuO+/H4vm46jh02J4LixERCLongt8Lk4lKGAAIIIIAAAgg0ESBx0ESJOgggsCgCurpAT0ZQ4mDQSZU6N+gkTduHtW8SQ3V88Xi32pMN7rQkwd3W1x328kvCNYfBB20SOiUItNT79zs33FCVn2/JBSUN9Cot9balen0sn4tNPHl3u6bx6m39++Bx6ut6YiG3n2PqjaY+v8kmMazHV7ybLGmgJ3L8ytR3R4kE/W9FS7ylQU9BqD8h4Yx169Lpa9dO7YEVAggggNZqSfkAAEAASURBVAACCCAwWIDEwWAftiKAwAIL+K0Ivs7tPnfSFevVT7LiNr0f1t7rl+LcYcmCZ9qcBTfYoxev2by5undciYIdmseg4USHLzvmmKSkgS9N++T1Wc9dYBTzeqKgSdvYpvRd0ihiUuE9Z501a2C+L8X7/po16Rr7ru20J3B4kuqDt99eTXz4/he9aPq7rSSCX5EQb2Ng3oJZvBQggAACCCCAQAMBEgcNkKiCAALzL6BEQS5Z4CdNuR4MOhlT/UFtY7xhceqxTrGJ6jTR4fP0sr/2fmjARIelPsSkQexLm+/3WBKjL8uqVavmdail4xh3GhMFKh/WxusP+v4pqXCHfceesIlA77FkwA9tokO/smWH3RYTE1UeJyYi1IcvW2Lhx084IV140klVd3VrQ+6WhjgW3iOAAAIIIIAAAlGAxEHU4D0CCCyogCcKfK2dDzrZ8hOjXCcHtfP6g9p7nVycB+wvvZpQ7mg7CfMl1ms62Zy3HXXdpwTAqDZefy5GbSUd4nfC+1Vfe7JA5YPqez19Z/VEDi1X2mM8n7KT/mETHcbvuZIIFzzjGVV731/ploZ4ZULVgH8QQAABBBBAAIEpARIHfBUQQGDBBeLVBX4yU+9EPPmpbyu18XqD2nqdYTFUT3VutBO379h6pf1Fe/WBB6a1dg/5c+0v+Ud7oMxakxzqNeiqgrmc6GZ2SdEcBEY5FnNNMgz63nmyQEOp19O2d51ySjWhZilRFdt4rPr/FvxqBK9bv6VByQN/zYGUpggggAACCCDQMQESBx07oAwHgaUqoGSBPx3BT1rqfa2f5Gh7qa63zbXxbU3aD6pzryUJ/sUuEV9pJ1M6Ydy9c2fabidw99r74+wJCnZzevpR3Q5gyQW917wFmuTwBLsPXcsoJ6RVg5b+GWbW0m6WdBg/cW67k02P6agJhu+vXp2ute/a/1G75cLHMeiY6jv3C2ecMWOosb7HUIX4vxdPIqhc9etJBJVzS4MUWBBAAAEEEECAxAHfAQQQmDcBTxT4rQjxZMZ3Gk9kvCxXT9tydb2N1qV2XmfYdq/n6z32RIdtjzySVilx4C9LGKy012Z76S+/P24nXC+1hMFLj568BuFuO/lraxm1v033O8yxaZzFrhdPfOt9mYtdPNGux236eViCQYmF2w8/PN1lEx1utokOd9jTETTZ4X+x178//fTp3cRxqF9KEvzQ5jbQ4k/kOPWww6rP8bhGmxhDFX18g+oriaCl/pQGbmeoWPgHAQQQQACB3gmQOOjdIWfACMy/wLBbEeIJi/emfnLj5aPU9TalWL69yfrbdnXBV26+uUoOKFGwnyUOqisPQgLhLVMnbJ40UNwT7XaGR+yKgyPsZLC0zKV/OY/SfuZavpD7in2NJ72xvP6+jf7l9jXK8fGT8HrfSp83HXFEetKSS/daUurue++tkgWa7DA+kePDN92U3vM8Tb05c1G/TjzooOr1iuOPn94Y++v9qdvEcY5S3+syL8I0N28QQAABBBDopQCJg14edgaNQPsCflWBr/2EI+6pfjKjbXOpV2of9zns/cC/DNvtB9U+7CRvp73ssQ9phf2leD+7rFxXIKSpxEHch8ZzmNXTNIr/3wEHpDWWSDjCkhDPsddRipFZci6ZasWiubYvBl6kDfM5nngCreGNs68YI/f9zbH5Cf1JdgWLlo333ZeesqsM9tj3wucs8HWuvZf5vBlxvx5bdUrlcZyl/sc4ufoem1sa/GiwRgABBBBAoD8CJA76c6wZKQLzIrAYVxf4CcwoAxqYIGgayBIJey0xsMuuJjjWLhsf1I//ZvMjrLS5D3RiuM2SDffZa4UlG46xfa21y8APt8vNn2FJhfoST9jq20b9PKh/o8ZajvXjibD3f1zfeLLdNEZsUz8Wv3rccekye7RiKVkQv6/D5kuox/Zxl8pj/0t99BhyG1SfWxr8m8UaAQQQQACBbguQOOj28WV0CMyLgF9VoHX95MR3GE82vCxXt2m9XFuPW1/Hk676tiafta9cv9T2W5/7XHqGXUa+2d5v/pu/Se9617uyIZ+yvyxP39pgtzroCgVdrXCvJRzus88r165NbzS/Ycso4x4Wy7eXxubbl9s6nvzGvs/FLp44K+YoZrfYEzVusITR/167wiT2s9Q3Pb3jaZtw87LvfW96KLo94d+eeur051ESCXEccZ9eXh+X9zHW1Y6b1ueWhunDxBsEEEAAAQQ6JUDioFOHk8EgMH8CCzHRYf1kRaPJleVGOWqyoGnc3L7+5r3vnS7+1re+lS6++OL04he/eLpMb560xIHmRZieVNHea66EVbrNwdb/1q5G0O0Moy71E71R2zepvxD7aNKPWMdPaGOZv2+rv3Efo34/7j7qqLTJrkS5xxIGO+1KAs1b8Ac20eHbTjzRHrgxecvLoH6+xNofYBMffuP++9PTQybYjN/1NpMI8ox99H6rPHp4EqFU3+tyS4OEWBBAAAEEEOiGAImDbhxHRoHAvAnoqgJPGvgJQdxZPNGI5U3r5urlymJsfx9PoLystC7FLPXfzqBKoWaUf/Ob30x6ffazn62SB76fvfZX41162UmgLkevEgZKJExdfZBsRn0txf3bti1Wf53d8jBoGdR+ULu2t/m4m8aNJ59N2szXOOPJ8aj7uNdO9jXR4fV268nTDz1UTXSoJyNst9tQdIuKEkVaYty4vxevW1dtV9JA5Q/a92TUJf5vYJQkgvbjxyAeOy/T9lK/x6nPLQ0SZUEAAQQQQGD5CpA4WL7Hjp4jMG8C8VYE7SSeKPhO40mFl82lXmk/Hjuu48lSLK+/b9oftauPp1naYN8e3/SmN6UXvehF6TOf+UxV+I5nPSv97T33pM12YqnYu+3kUnMj6MTs1047bdb+6n34W5v/QJetr7D10XbrhB4NeZYlItbaCemwJTfuYW0Wevti9TGeGGvM9ePe1EEn+sdbskDL1+zqEj0VQcfX5yzwpMHl9mSOi+3xir7fuD8lDHypygckq+TlMbxNfR3/dzEsiaC28Rh47FimOl4e+61yT4A0qe911c7rc0uDNFgQQAABBBBYPgIkDpbPsaKnCMy7gBIGnjTwH/hxp/WTB23L1VN507ql9ooRl3hSFMvr70vxcv3JlXn751rgK+vB7fNn3v/+9Byb5yC36LaF5zznOenTn/50lUT4X044oap2t7les3Vr9f4eO7n82G23VcmDXAyV/bn+Wm0nkbo6QcmDamJFSzh8XVcg6JYHKzvdYjZJIiiej0nv+7b4ia+Pey4WMVb87jy5ZUsV3pMGvi+tlTTQEvfrcWKMqtKQf3IxSk3q/3sZlkgoxS6Vx77nEgPqV2mcXt9j129p2N++4/4qjY9yBBBAAAEEEFhYARIHC+vN3hBYcgKeKPC1/5iPHY0nCV4+l3q5th63vq6fANW3++dczFy/c2Wltn7S5/vQWu3f8Y53pI997GOxeMb7N7/5zVUd1dOipyfopf0ocaAlt89qg/2zzf6CPT03wtStDdVEi5YwuN9eSiq8wOLEORIU94Q1azwE6ymBQc5NkG63R27utqtGnmNXe9Rj+YmxriD5/dtvz4bz75ufLKtSjOMxVD7oKpd6nFIMxckt8X9HoyQRFMv7WNqn9011G40zXFlRr19PIijmQTY5pBIJLAgggAACCCCweAIkDhbPnj0jsGgCPmdBG8kCDSKeOPig4knGoDLfFtfxJCeW19/n9pHri9rlypu2r7cdljjQ/pRY0BUIv/Zrv5Ze+MIXqqha/OQ+t2+vs90ue9fJmq4sWHXAAdPzIngy4We8Ylh73FA08ttBfRo52AI18JPaNnd3zzHHpM02t8R99tpp8xVo3oIzLfGjJe6v5PXCI4+skjjxcZv175CfMM+IYcd72BLjZGPU+piLF//3NSyJoPaxjz7+WKY6Xp7rXz1G0/rMiyApFgQQQAABBJaGAImDpXEc6AUCCyKgRIEnDeo//NWB+KM/dqhp3Vy9XFmM7e/jyYyXlda5mLm+58raaquEwO///u+XuliVX3PNNektb3lLevvb3169BlYOG5UEuNfmRNC8Brvt5FUniLpnXk9p0JUH6Ygjqtq5sWjDFXbv/Go70T3UTnqfZcf8SIvV1pIzbSt2PY6fGNfL4+eSQazT5P2T9kSDByxZ8127KuRpzVlgyYKdZqckzh47Bn9sJ/X/58knzziJVlw/YX7B1DHR2su0PXrF8eTKS2NR3Vz9XJn2GePEvmhbfan/725YIqEUO1ce+6f9xvGPUt/rMi9C/ejxGQEEEEAAgYUTIHGwcNbsCYFFEfCrCnztP8JjZ+o/8LWtab1S3Vz7uE9/Xz9x8fL6Ohcv1+9cWa6t4ufq5spy7XP16n32z3/wB3+QlERQAiFefeDbtY77eN3RR1eJA5V/+7HHqvdPWxJhp71+2Wbij3VVxxf16fOWMFhpty3stnkSttvJ7kN2QrzG4h1qJ8HPtETCU48+umxuaRjF2A3GWeuE9mCzuc4SB0oY7LD31USHVq5FV36UFj8Wz7dbGvwk3cu8jZfXx+Mn0tPl4RJ+b5tbT9e3jbNiTDXw8tgX70cuppfF/z22mURQ/Fy/VV7qY66+1+WWBsmxIIAAAgggsHACK+z/mAfdVrlwPWFPCCDQqoASBW0kC9Sp3H8m/Ad87HSuLG739/HkxMty61K8XH9yZbn2uXq5slzbetnZZ5+d6/bAsre97W1Jr0FLrj+6AuE6++u31m99xjOKzf/SEgb+yEfd2qArFXTCuMJOfnUCrAn8Xn3//WmLvdejIo9bvboYq+sb/OQ6jvPjmzZNPxkhluu9rjhoupRO0kvl3hc9hUNP44hLfa6NYTFiW7332PXyUpx6PX0elkSotynFzpWP2r9h9TUfwqB5ETZseCRdeuk/alRVt9evX5euuuol9SHwGQEEEEAAAQSCAFccBAzeIrDcBTxR4Ov6ia7GlzspnUu9XNuS41wSBrl+58py/cnVy5Wp37n2uTLV1RUEuppglOUP//AP07/8y7+kX/3VX00veMELBjaNfdQJ/r+xqw20xPJ6gG329Ib9rG41H4KSBnYS5e+1fq0lH5QtXmtXL9iMcwNj1WMv98932dwDSpacMXXrRnT0k9HckxF83PF7kDsB9npax7r67PVjuZdpu/dFt6noUZ5x8W3exxhD9TyO11OZ19X7UnmM4zFUP7fU/7c7LJFQip0rj/3Tvr3vsa7KvY/D6nNLg7RYEEAAAQQQaFeAxEG7nkRDYMEFfM6CNpIF6nz9R7nK6j/gS2Uqry/1E476dv+c20euL6qfK2/avmnbXDzvq6/HSRyorRIHnjxQAmHYkutzqY1uadhhVx3oBFhXGWhehCpxMLWOT2Kox9h26KHpIEssPGInr0dMPf2hXme5fX7AHom5zW4/uMXG7/MWnG4uvvhJqhv/ktX/ZPiL/3lmokW3IsQlfj/8hDZur7/P1Y9lqu9xvC8q8/7pfak8xsnFiHFijFiei6HtpSX+77rNJIL2F/sYx1/qY66+183d0qD/ZrIggAACCCCAwGgCJA5G86I2AktGQIkCTxr4j+TYufhjOpY3rZurlyuLsf19PKnwstI6FzPX91xZ223Vx1zMXN+9nm470FUE4yx/9Ed/VCUQ/viP/7hRc9/noMo/aX9Vv8FOlO+z5MFee4TgLr3sr+xKJFxkTwvwJed5oM2n8A9WR7c06AT0SDvBOtXKDl9mSYSn7GT/QUsU/MDmKHja5ivQUxGUNPCJDj9l43vL8cdXFNFBJ6lKGujqjuPtaRaeNPCT11hXjb28flz85N2t6+tYP9bNlZf2WSrPxdD+Y33vd6k8xlCd2Ed9ri/xf++jJBEUx2PHfXqZtpf6PU59JRG07NplV9uwIIAAAggggMBIAsxxMBIXlRFYXAG/qsDX8cez9yz+0PaypvVUP1c3V+ax4zqeQMTy+vtcvFy/c2W5toqfq5sry7XPlX3fTi7PspPuuKieJius//X53HPPjdXGeq/kw4/92I8NbZsbU6mRkgdKItwf/sIakwfezmN+0a40qG5tsFscqjkRLHlQzYtgFkdarKfN4zRLWB1uiYiluNxvcz8cu3lzdUL/BRuLPxUhTnTo/X7zscdWb+NJqgpkpsRBPLn2NlqPWl6PH2PV35fqlsrb6EsbMerj8M/Dkghez9elcZbKS33P1Y91P/zhXen3fu9a2+3klSfnn39Y+uIXz67mRVBfNEcCCwIIIIAAAgjMFOCKg5kefEJgSQooUdBGskCD85PEONDciXOuLLbx902TBaqfi5nrT66s7ba5eD4mX//RXXfNOlH0pEHso2430NUDc1l05cKv/MqvVK8mceL+S/WPtROg10w9JlDJAyURBrXbZk9cmDGxorXXZyUPHrLXCvt8mH0Xl+KMujoxVNJAi8b4OktufOaBByZv2Zh6OkLOqf498Mki605+4hnLvcz36fFjeYyfO6H1NlrHuvrs9UvlbfQlxtA+ve+x3Mu0PfbF+6fy3FL/b8OwREIpdql8lD7GuvYNmdFdbavf0qDkgb9mVOYDAggggAACPRUgcdDTA8+wl76AJwp8HX88e+9n/hieLJ1LvVxb31d9XT8pqG/3z7mYuX7nypq21b6ats/F9L7GdS6e76e+rcmVAjF26f2f/MmfpGuvvbZKHjz/+c8vVZsur/djekPmzTGWAHj14Ydnnbz6dksc6EqDOCdCnFjxJ23+g/ocCdvt3v81dhXGYi1+8lq30MnuoIkOY/0mJ8a5+rFM4/c4sdzLtL3+3fO+a1tuifVj3Vx53Kdi+X5juZdpey6GynP1Y5nqeJwYQ+Wxj/pcX+J/M0ZJIiiOx4779DJtj330/qm8VF/bcovq15MIqjfoKQ25OJQhgAACCCDQNQESB107ooxn2Qt04eqC+GPdD0j8Ye9lWufKm7Zv2jYXL/bB38+oZ5fpD1rivvV0hDauOtD+vv3tb1dPa/jlX/7lKoEQ95Prz4w+5yqMUmYn23vtxGm33Quuy/t1YqarDVbZ/f668mBvxuQASzb432+/duqp6Qhru8cmaDznwQdH2XOjultOOqma6PD2qdsQNG/B6/V0iKmlfiJ59kEHpRvtCglf9FnLOQcfXH3v/ASzbuzl0bYe22N6XX2Ocbw8lqmOl+t9Kb621ZdS3VJ53K/vM5YpvpePEkPtYhyPofJSHG2rL6MkEdQ2FzuWqY4fo2H9i9vVTouX1cezb16EyQkVhz3qcTIa/yKAAAIIINA9ARIH3TumjGgZCvhVBVrXfwz7cPyHrX/WOle3ab1c2xg7vo8/8mN5/X0uZq4/ubK226pvuZj1Phfr2eXLpSX239/7utRm1PKPf/zjVRJBCQRdfTBoLG3t++dsYsUH7ET8pqk5DPReSYDddpvDz69dO3AIVx59dFph7bbolgZLNHztlFPS6gMPTAdZYuFke+kWh3GWHXZFw0M254CeirDH4u+0144tW9J2u8phj/Vr71FHZU+AtS8lDXS7hl6eNIgnhtFtWHndf9hJ6iix1dcY32OrPLfEutru9WO5l2n7KH2JMWLsGEPl7hXLvUzbY5zYF22rL/X/voxyNUKMndtn7J/2G/vo/Yh14vtY32MrkeBPZfAkgurpPQsCCCCAAAJdFmByxC4fXca2pAX8iQieNPAfprHT9R+x2ta0Xqlurn3cp7+v/5j38vo6Fy/X71yZYs2lfa5trqze59J+Ve79/Ff7C/s/3XnnrBMNzbSvl9eLsdu6ZSHG1Pu3vvWt1cvLc/v2bW2ulTjQokSC3v87u9WhtHxJVyXYyZPf2rDS/HSrQDW5oq2PsEkHnzv1mEPNt6CT+UHLlpNPTut++MP0j/Z4xB2WANCTETTZoa4yqJ/8KeHhS9wW9xNPML2u1rG+l+fKSnVVPkrsUhzt82EzWWs2cSnFjnXi+1L9UnlurLky7WOUGKqfi1OKofq5ZVgSIbYpxa6Xf+Qje9Jll11nTScnR3zpSw9Jf/3XZ06Hqtf3DbnxaJvqe+KAWxpcizUCCCCAQNcEuOKga0eU8Sx5gS7ciiDk3Al67oQ2V9Z221y80hchVzfXx9L1Bqob68d4mtxQ8xS0vXziE5+orj5QAuG8884bGD72Z2DFBhuPmro14UK7tF9LHHe9+batW9N+dmWAEgf+ZAZPJCiJ8CP33jt9S4PmWxgUS7GVNNDy45Zs+K4lHW5SwsD6Uzp5qyrbPzFuTE5El3hiGOt77FimuMPKR4mteDH+jNj23fq0XU2hx2fGRY+GfJ1d0aEl9j3W8felvpTKi33xgLb2Po4SQ81zsWMM1Rk2npjAHJZEKMWul2u/g5ZYP/YvNx7FUX1uaRgkyjYEEEAAgS4IkDjowlFkDEtewK8q8HX8Yeqdjz9KvWwu9RQj195jx3X8cR7L6+9z8XL9zpU1bat9Nm2fi1nvsz7n6uX2obqlcm2LSy7mfCUOtN/rrrsuXXzxxemXfumXZlx94H3K9afpWDzGsPWgeLptYLed4OskU4kCTyJUT2WwZMKgtnG/fpIa6z/XEhLfqZ1Me5t4tYGX+TrGUJnHjlZNTgxjHI+heLnyGFt1PH6ubj2GPg9bYnyPXWoT66qO14/lXqbtTfro448xYuxRYqhdjBP7om31pf7fqVESCTF27KPvo142yjgVo16fWxpcljUCCCCAQFcESBx05UgyjiUp0IWrC+IP+4hc/6GtbbmyXPtcvVxZrm2uLPbL35fq5faTK7PBeKgZa9WNsett5zN5oI588pOfrJIIH/3oR2f0wztZ74/Kc2Vef5T1oDg/Yif3D+7ZU72ettsadiiJoCsEbM6D/83mOshrTu7dT7r0Ke7Dy2OZ9/e59ld4LX4s4omh16mvYxyP7e29rseJdbXN68dyL9P2UnmMPzS2AhUWjx/3mYtdaF4V5+rHMlXK9THu0/uhurE8xsnFiPVjjFiei6HtpSUmEkZJIuTiad/eb22PfRw2zlJ9H4+SCHopPrc05PQpQwABBBBYDgIkDpbDUaKPy0rAryrQ2n841gcQf5T6tlzdpvVybT1ufR1/bNe3xc+5mLn+5Mrm0lZ9yLXPlcX++vtcvVwfc2WKUSr3+L7O1VOZJjOcj9sVfL9a6+qD888/P/3TP/3TdHGpP9MVpt7k6tXrlD4PanuWXVWgl5IHWr5vVyDo/V69CkkYPyGL271MMWK5PmtRwsCTBpMlk//Wj3s8CYz1/H2MHfcZ48QYufqxTHE9Tiz3Mm0fFju2U31fYnl8r+0eP8ZWeey7PteXWD/WzZWX9lkqz8XQ/mN973epPMZQndhHfa4v8b9rw5IIsR8xTtxn3F+sH/s9Tn1uaYjivEcAAQQQWE4CJA6W09Gir0tWoM2JDjXI+EPVBx1/pA4q821xHX9Ux/L6+9w+cn3JlSnWXNrn2ubK6n0u7VfluX42LTvd/kJ4+nHHpU/ff//0Ll9zxBHpOPvLeowR33tFJQ/0VIT5XvToxvqcB7n+NC0b1N+mx2Kd3aqg5eXmpOUh+0vuRrvN4BVTVwjEE7KH7WqEI6cmYPQTMp+Q8WjbpsXL19ujFL1M4/HyODYvU7vY37hPbasvMYa2eZxSjFjf66pdrjyWNYmtOrklxhm2T7Uv9T0Xu1S3VD5KX2IM7duPxSgx1C7G8Rgqzy3xv3fDkghqr754fI/tnz2+l8d+a5sfiyb1va7aeX1uaZAGCwIIIIDAchAgcbAcjhJ9XLICuqrAkwb+QzB2tv4j07c1rZurlyvzuHEdfzzH8tz7XMxc33NlbbfNxcv1WWW5urk+Ni3LxXvzscfOOCH0vjSN6fXnY3399ddXiYNcX7S/XHmuLDfuXH9zbXP1YpkSCZ40UHnc1x12C8NXbfLDY+xKBV90lcIFVu6L79MnaqyX67OfkHldr+PlcZ9+Auh1cusYJxdDbTxOrq62j1Lu/dtrSZbSonjel1FiK57H13vvt97nllhX271+qbyNvsQY2uewcca+eP/ULrfU/zs4LJFQil0qj333fqsfufqxrup4fa9bv6VBtzX4S/VZEEAAAQQQWEwBEgeLqc++l6VAvBVBA/AffXEw9R+Ic61Xah/36e/rP5S9vL5u2u+mY1H8XN1cWW7fubJ6n/W5VC+3n6ZluZhN2+bqaQLDhbjiQPuu77/+WWa5styYS3Vz7VV32OInRvX2Xu7t/fYG/6z6sY3Xz5WpzSjl9XEPO/EsxY5xPEasq37l+l0q97raXlpifK8fy9QuV+5l2h77rc/ed73PLbF+rJsrn0tfYuwYx/sey9RPL4/9UHmMo8/1Rf99tK9Xo6UUO5bH/cU+ev+0o3Hq15MIisOjHqXAggACCCCwWAIkDhZLnv0uOwElDDxpEH8I+kDij0Yvy9XTtqZ1S+09vq/nkiwo9adpH3P1cmW5seTKfExxXaqX20/TslzMXNtxffToRD1GcSGWUfrddNy5mLmy+vjiCVOsXyqvt9fn2K7+2ePU6+TKvawUQ+XRI54Ealt9Ke2zFCPWH9YX1R10Pqt9xP6NGjuOJfal1PdY39+X6pbKc32MZYrrfRklhtrFOB5D5aU42lZaYqxSHZXnYscy1fFjFGOW+te0PvMiSIoFAQQQQGCxBUgcLPYRYP9LWsATBb6u/0hU5+MPRB/MXOrl2nrc+nouCYNcv3Nluf7k6uXK1N9c+1xZfWyltrn95MrUPlee23euXtOyUjxddTDfiQPdqlDvZ/1zybHkk2vfpMxPjup1c+Vepj6UlujqJ2OqG+PHOLnyWKa2Xj+We5m2l/apbbklFyfGUBvve66utsdyfR60zCV2HGfcZyyP8b3fpf7Euqrj9WO5l2l7aZ+58hgjxo51Ve59j+Vepu0xTuyLttWX+n9L27ylIfZP+x3Wx1J9Hw/zItSPHp8RQAABBBZCgMTBQiizj2Ul4HMWtJEs0MDrPwJV5j8A9d6XXJlvi+v6D9y4Lb7Pxcv1RW1y5U3bN22bixf76+9L9XL7aVqWi9m0ba6e+tokppIHenzifC2/+Iu/WIXO9THXP1XO1Z1LWdWBWtx4YhRje7mX+dpj+Frlenn9+lj8JDC297qKMUp5rKu2Hqe0T9XJLTGOx1C9GCfXb9Xx+odv355+9pBD0mdtMklfLjrmmOqtx/e6Khwltrf3uB4nlntZKba3za1zfYllajNs/KP2JVc/lmmfPqaZfVmpTQOX+N/ZUZIICurjjPv0Mm2PffT+qXyU+l6XWxokx4IAAgggsBACJA4WQpl9LAsBJQo8aeA/ymLH44+9WN60bq5erizG9vfxR6yXlda5mLm+58rabpuLNx/9zo1F+8ntP1e3aVnTeD5G3a4wX4kDJQ3OOeecGSchcx2z2je18DHm1vUYfnIUy70s197LSvXjcfATslhX7T1+k3Kvq3axfizP7VP1c8uoMer133T00dNhfZv3xT97BS8v9S/W97pqmyuPZarj9WNslbu53ueWWD/WLZXH/fo+Y5n24eWjxFC7GMdjxDLVGbbE//4OSyIoVq6PsUx13CX2xftXiqHyQfW5pUFCLAgggAAC8yVA4mC+ZIm7LAT8qgJf13/caRDxh5oPqmk91c/VzZV57LiOP1hjef19Ll6u37myXFvFz9XNleXa58rqfdbnUr3cfpqW5WI2bZur1zRebnwqe8tb3pI+9alPlTaPVX7uueemN7/5zdNtc33Uxtx45lI2vcMR38R9+slRLKuH0za9vK62x/qxPI7dT8YG1c/FiWVq6/FjuZdpe2mf2lZfYgxt8zilGLG+11W7UcpjbLV1l1FiDNpnjO+xVT+3xLra7vVjuZdp+yh9jDFi7GEx9u6dfGSo2vjisWJffFtc1/+bPCyR4HEVI8bOlcd+q74f/1g3xhlWn1sapMWCAAIIINCWAImDtiSJs6wElChoI1mgQdd/vKms/kOvVKby+lL/YVrfHj/n9pPrT66s7ba5eLGv8X2ubq6PTcuaxlMf5iNmHJve5/pTrzOXz540yO2n6fia1ptLP3Nt437P3ro13VGr9IbVq6dPmGJdVfMTqVjuZdpe9/ATtVL9Ucpj3diXuE/fn7aXlhjH+x5jqJ3HiXVV7vVjuZdpe6k8xl+I2OpLacn1JZapXa6PbY9zlP6V6qo8/ve6zSSCYo9yPEv13ZZbGiTEggACCCAwFwESB3PRo+2yEvBEga/9B1UcRPyh5uVzqZdr63Hr6/gDtL4tfs7FzPU7VzaXtupDrn2uLPbX3+fq5fqo+rnyXFnTmLm2ubIY77YDD0zPsnvN6/VusrLn2bbcEtv7dl1xoKXNqw6uu+66dPbZZ/suqnW9nyqcS9mM4PP0Qf17wwEHVNGbnBjG8Xj9WKZAXq738XjkTkZj/RgnxhilPO5PsX2fep9bSrFjnBgjVz+WaR/e91juZdreduy4zxhb5bHv+lxfYv1YN1cex6M4PqZSeS6G2k3Wn/38ihjHY8cYahv7qM/1Jf43fJQkguJ47LhPL9P2XP9UPk59bmmQHAsCCCCAwKgCJA5GFaP+shPowtUF8cehH4D4Q9LLtM6VN23ftG0uXuyDvy/Vy+2naVkuZtO2uXrqay6mQab/tmXL9AmKj+lH16zxt9U617a0nxkN5/DhM5/5TNX6oosumhUlt++mZbOCLWBBvY9+8tak3Ouqu7F+LI/HadgJWYyhmB6nSbnXVbvSPrWtvpRil2LE+nGfufJYpv16/TZiK16M77FVXoqvbfWlVLdUXtpnrjzGmNzvzMkRYxttj5/HHc8oSQTtM/bRv5+xTHW8PPZP5d7HJvW9rtp5fW5pkAYLAggggMAwARIHw4TYviwFlCwYNNGhBlX/8aUy/yGl9740rZdr6zHq6/ijsr4tfs7FzPUnV9Z2W/UrFzP219/n6uX62LSstO+m7XP1mvbRxxTXo7TVvjWRYZtXHKgvSh5ocsTcBIne19y4c2VefymtYz/jyU6uPJZpDF4/lnuZttePX+6ELNYvxcmVx7LYl7hP35+2l5YYx/sSY6idx8nV1fZRypvEVkzvyyix1S7G936rPLfEutru9Uvlo/Yl1vf918uGjTP2xfvnserr+n/vR7kaIcYu7TP23futPuTqx7qq4/W9bv2Whv333z/5S/VZEEAAAQT6KzAz7d5fB0beAQG/suChhx6q5i/QDyD/MeTD048mf3mZ6vjLy7xO/UdWvZ7q58o8Tlzrx6O/Ynn9vcfT2pdcf3JlubaK4XU9XqnM28d6Xhb7E7f7+1K9pvvO1cvF9Hpa+9K0LBdPMby9xxu0jvsttfV4sa4/OnFQ7FG3KXkQ9xH7E8tz/Rl1X4tZv9T/UcpjXb2PS+57Uaofy2OMUcrj/vR+2FKKHeN4jFhX731pUu51tc7FVnmMo89aYllpn5M1J/+NsZuMP9YvxfHyUfvi7er9iHG8TiyL44z9q8fxtnHt/z9QTyjEOv6+FDuWe12tYx9j+Tj19f+hTzzxRHr00UenX0rIsyCAAAII9FOAKw76edw7NWpPGGhQuR9t8QeeDzxXT9ua1i219/i+bvLD0OvmYub6kytru20unvezvs7VzfWxaVnTeOrHfMSsjy/3uel+Yx+VOPjTP/3TXLixy2644YbqyoM3velN2Ri5fmYrLqPC+pj8L6ax3Ms0rFHL4/cv/rU3FyeWaV++31juZYP6Utqn2tSXGFvbPH4pRqzvddUuVx7LFiq29lPqu7bVl1LdUnkck48/5FNmhB8lhhrmYqu8FEfb6kv8/4hhVyKobS52LFMd/9426V/T+koiaPHEga5COOigg6qrEaoN/IMAAggg0HkBEgedP8TdHKCSBVp8Xf/hpG3xR5M+a5lLvVL7KnDtn/hjsLZpxsem/ZnLWLTDpu1z/ZnR4akPuXq5fZT2navbNGauba6sabzc+Er9ztXN7btUNh/Jg89+9rNVtzx5kNt3rt9dKYvj3XdiuO8v7RpnrtzLtD0XQ+XxO+QnY4Pq5+LEMrX1/cZyL3vM5s84bPt2VasW3+dXH388verQQ714xjoXJ/ZblT1Orq62j1I+n7HVlxjf+63y3BLrarvXj+Vepu37xjnz+xG3+bGIMbTd4+yLodLBx1PbYxyPofLcUv//jWGJhFLsXHmp3+rHKPW9rhIJ9SSCYimhwIIAAggg0E0BEgfdPK6dHVUXri7wH17xINV/1Pm2XHnT9k3b5uL5/uO6VC+3n6ZluZhN26pvubpNY8ax+ftcW98W17n9NinTYxSHXXXgEx76BIhxv6X3Sh5o/548KNXrenk8Bn4CqDHnymOZ6nj9Unn9u+EngbG+xyjts1TuMXQ6+z927ky77MkdcTlmv/2qkzvfZ9wW33sclcW+xL57jFg31o/lMUapfNzYTfaZi612pSVXP5apnY8/xohji+/j+GOcGCNXP5ZpPx6nFCP2Jb6PiYRRkgiK4X0s7TP20fundqPU97pKIuilfXrigKsRpMmCAAIIdEuAxEG3jmcnR+NXFWjtP1TqA40/gnxbrm7Term2Hre+jj/u6tvi51zMXH9yZW23Vb9yMWN//X2uXq6PTctK+27aPlevaR99THGda6vts/8mOdmqvv/656pt5lpo389ll12Wrr/++slg9q9uN9CiSQ49aaCYevnVBFWFIf/82Z/9WRWj/pjGIc06u7l+XPzkKJZ7mRBGLffjqbZ+khZjqNzjNyn3umpXWnL7LNWN+4yxSzFy9WOZ9uNxSuWjxFa8GGeU2Grr5nqfW0p9UXncr7eNZaP0JfYjF0Pxc+Wxf6oT4+hzfYn/PzMsiaC2Mb7HjmWq4+W5/pViqHxQfW5pkBALAggg0E0BEgfdPK7LflT+RARPGtR/8GiA8ceLDzhXb5S6pfYe39fxR5yX5da5eLl+58oUby7tc21zZU37rXq5fjYty+27adtcvVy8Uh+bjjG3n1zbXL1cWa6PShDkTu7VPsZQEkFJhRtvvDHXhWzZe97znnTFFVdkty3FwpvWrk3Pe/jhWV3bYhbrVqyYVT6Xgmg77MQwV1f7LpXH4+wnY4Pq5+KoLJbHsfo277e2lfYZ2/n7elyPU4oR63tdxRqlPMZWW3cZJcagfcb4Hlv1c0usm9teLxulj/XY3pdRYmj/MY7HqPfLP9f//2dYIqEUO1ce+639+fGPdVXufRxWn1sapMWCAAIIdEOAxEE3jmNnRtGFWxF0MOo/slRW/4FVKmu7bS6e9p1bcnWb9jtXLxdP+83VbVqWi5lrO+fx2YlcbvmR1atnFOf23bSPubax7Pd+7/fSa17zmhn7G/ZByYMPf/jDw6otme1/Z5c4r7BLnOPyMrs0fz6XaOwnRtpfrjyWqY7Xj+Vepu31Y587wYr1Yxy1H7TU63qcuE/fX9M4uRhq63FK+4zlHkPtSuW5Psa6autxYrmXjRpb9UtLjO91Yv9Ulhv/uH2p78/jlMpjX7wf3s/cOiYS2kwiaF+xj95vlZf6mKvvdbmlQXIsCCCAwPIVmN9fZ8vXhZ4voIBfVeBr/5ERuxB/jHj5XOopRq69x47r+KMsltff5+Ll+p0ra9pW+2zaPhez3md9ztXL7UN1c+X/P3tvGm3ZVV0H7+pVjUpCJQkBao2FEKAqQYwbgYMCfFWM2MjDzRjBiYwhDqEzwbFhJP4V8ivJEHZssI1J4gG4yPCXkXzDX8A/XIXxR4wtJAuEGkxnCdFICPWUVKWmJKFvz3vffDXfrrnv3fve+/q1xrjvnDP3XHOtve57956z3mkc1qrpfB3WqoccnbX6a+wLjx1LF+7ePZjzl/N159owUB7juRgYc9we7Oqrr064DKHVcIYCLnFY7/c7aK2Xvhd6YNSDKxdxVUd/L3gQWOOXuM6BY6pNTGNqPOCMqVq67jQwrjqq4fiKwZc5Kk5sMbQ1puYNXHPHtjPkCT/lqg5xnQ90OKca7jTgp3xq1HDVAIe5YN2Zfl/1NBGgRW2NSQzjtbwn4cclDahoWFQgKhAVWF0ViMbB6nq/1lS2a+HsAt1h0jdHd7CIO8z5O57DnK/DGF+XNZ6L04o5zVZf5Oa4rZo6N663+rq4JcamQYkj1izjQK+MwQZAT/OAXPpCN2x8Bcra86CuBScXUZSvuP6u2AOyylkumrlqA6e+4sQwXouJsdJUA2PUqWkon1z4OVyxWWvXYgKv5Y6x0mrcGq5zGjd/1UBcvv89GvBTHWoAd9bTRIC/01YMHMbUvIFz/i18cuFHflzSgGqERQWiAlGBlV2BaBys7PdnzWXHswqw5A5DOclyhwTjjtvKc75lTG7rjhYxt3SaLh+HTeOLXJy/w2adN/RmPZ9p9FrnV8vbxW7FXL2d77SxOUc2ANgQID5qSS59R3FjzFdA31M92HG4YlAjX3FiGC9/h3hAhjFnNR2HKwYtxtWY4+LBT3WcBjjUcdyaRg3X/GrawJlLb0zqqx/0nJHLMc6zhqsm84NvDVcdaisXvtRRnBjGnQZwZ+V3W8/ZCMwPurWYPTkqF5qcE7XLSxrwpAa+wA+LCkQFogJRgeWpQDQOlqfu6ypq3Ohw+HZzp0jf/HIHCmMOc74OU22u13guTivmNFt9HQ+5tmpyXrps9XWxWzEXAzm0+rfydF66jgYALkPouVkimge1mzGqdqyPr4C+fzzQgVcPrlz4qg5+v34w4owD+JKvOsR6cyl/n/XgEFql1WKqDjWUCx3m2IKTCz+nDVx1yFcMHIcTw3iv1XJxeG8uTgP5qQ5zVwwc4qoBnO8F1p1pI6GniQAtamtMYhjXHJkf8En4ZRMBOvGoR1QhLCoQFYgKLH0FonGw9DVfNxFxVgGbBrrDwALozgUxLFu5jucw1ea67jQRqy2dpsvdYbP2dXqLkXfrXBDbcVsxNx/n6+bY4+s0W7HWOE6vpz5ujiWGmx4eOHCghEdu434Hq+lmiSMns0IGy/eaB0eKE0PKrfip+TNzlDkdxeDLuIoTw3gN199zPQiET2mqgTHq1zSUTy78HK7YYmm7/kwtd+RQWo1bw3VOnL9i0CfeowE/1aEG8JoOxkrT78NxTQT4Om3FwOHvUEt+rfy4LwIqFRYViApEBZavAtE4WL7ar8nIeikCJljuTADTHQlsw6bh1fwHwsUP3UEqhhZstubTOheIO67DXGyHLUh4bqPGc3FaMafZ6ut4rXpufsBa/V3sVszFQOxW/1YeNHsNTQA8OaHV4maJrZWanKfvNw/eFIOyw4lhnPyfAnf7dkALDL+TPBjDAPlYdzqKK7cF199/jQlfZ6rPXFQDPtRxXIz34LPSRlxnqs+8HQ+YcrFNvuLEMD7reUKTNa9p13KBb2nld+S4RkJN2+GaH+Iyb6z38MmN+yKgcmFRgahAVGDpKhCNg6Wr9ZqOhIYBmwb8UtcJlzsMGHM84K3cmj801ModIR3T9Zqey8dhzt/xHOZ8Hab56rrjujitWKseclgMTZ0b1lvzac0Fmo476zguBmJPY3v37k29T1qI+x1MU/E+X33P9cDI4YohCvk1vPz95AGp8qkBvVngtZjQd1aLqToub2gx95pGDe/XPvlRqz3abt7EXC6KgTfp/FWHGtBzuSsGDmtb0wDHmX5/9jQRoMUcazE1R+YHvx4+uXFJAyoXFhWICkQFFrcC0ThY3PquaXU2CrjkF7hOWncMiE/Dc77ULZe6w1OO6bbTdHk7bBpf5OD8Hab5ct3xXI4Og4bDp9GcRo9zKpezzqc1x1p9nH8rVs5tmm3e8JANgRYtcunb4rNcnD0bNixX6JnGLX83eHCkODEE7sX174MHaaoBTeq34OSOysXFBN+ZxlTtmobjK4YY1KnhLdplrtDCq0cbGqx5qcftWi41XOfUk4vm4TSQj8M1D3BUB9ul6XfquCYCfFWf2oqBQ9zlV9MAPooflzSgQmFRgahAVGD2FYjGwexruqYVec+CWTQLUCj98mfhyh0L4A4jX5e6Y6N4ue70XC7wc3irf6uv0ytzxnaN5+K0Yk6z1dfxannWuOU8Z52Pi+tiIA/HnQYr5zbLbTYA2BBo0QZ3Jd0s8bIHH0znbN6crsun4cNen++evpZNf5fGHRg6LmpTw/V3mgdjo/hORzH4uhyJYbwWE2Ol1bRrGsrXmD24aiMf1sXd4wDjPdrgqz61gTtTLsbJV5wYxntyUQ3V7tGAn+poLhgrrfyuHddIqGk7XPNGXL7/ygXOHMfx45IGVCssKhAViApMX4FoHExfw3WhgEYBmwbllzcKUH5xsyitXMdzGHV1We7A6Fi57jRd7g6btS9yc5plzjWey7EVWwxNNxeXzzTzg6/TbMVac4Ted3bvTuc9/PCCdIF/7emn0yWbNs3jLvb84CKvTPKkBdwf4dChQ4ucWbv8WbmWr5d6tnuubqb+3vDACDNyuGLgkK84MYyXv+fuAEv5NR2HK6a5aEzGw3jNVIe5qAb8qKNc4OQrTgzjNZz6Og4+jBh1uI0xYsorcWoDZ95Yr5njKwY/6swiF9WANudUwzUX5gG/mun38CybCIinOTJv4LUcHZ/cuKQBlQuLCkQFogKTVSAaB5PVbV148awCLvnFq5PXL2jirTzwHddh1Nal7qgoXq47PZe3w5wv9B3XYc7fYWXO2K7xXJxWzGm2+jqe00Pujgu8NOfvfKfBXAzk0aL5/z7++PwONnNH08D5cnypl5M8aQHNg3jSwlK/U/V4+vukB0Y9uHIRSXX0b4AHgTW+4qrRg2s85MKYWHdW01Yd1XB8xRCDuStOzOVAzPEVq2krrnkD19yxXZrylevwaXJRbdXRujhc80DuqlPOBdv63dzTRIAvtTUmMYy7/IBPwo9LGlC5sKhAVCAq0F6BaBy012rdMNEomEWzAAXTL3kWUL/gR2Ec06XukCju1l0cl4/DZu3r9FzOwBzX5diKteoh9mJoQletNZ/WXKDtuNPGOfn2acNZuFg6v+VY771ZYjxpYTnepbaY5e8XD+pacHIRSfmK69/FuAMy1YAmdVpwcuFXi4mx0mraNQ3la0yHn8BO/uum/riaIN8TOidqMgqnNjiqj+3Satwa3pOLaiAuc+nRgJ/qUAO4M/3OHtdEgL/TVgwcxtS8gfP9b+GTCz/y45IGVCMsKhAViArUKxCNg3pt1tUIGwVc8otUi1B+SWNsGp7z1Xi6rjsfipfrTtPl7bBWX8Rs9XeaZc7YdjwXA1yHO6xV0/k6rFUPOTpr9XexWzEXA7m0+juem8tKwya93wHmQd+VNqfIZ1gB/Z3Ugx2HKwZv8hUnhvHy78UdkCm/puNwxTQXjcl4GK+Z6jAX1YAfdRwX4zUcYzAdb9GGD3NRX2KlpuKqz7zBd6ZcjJNfw2eRi2ogJnNXnBjGNRfmB9xZ+T0+rpFQ067hPTkqF7lyTtQuL2nYku+/wpebW2BRgahAVGA9VCAaB+vhXR4xx7VwdgG/6HWa5U4Bxxze6t/q6/QYX5c1novTijnNVl/k5ritmjo3rrf6uritGGItSpzaHdQ4uRW2ZAOg92aJmAZ9V9iUIp2iAvo3wQMdUHpw5cJXdfTviAeBNb7iqtGDazzkwphYd1bTVh1qKBdazBG4+9NWPrnwc9rAHV8xcKijODGMqza2mTvWnSlfuQ7XmNBiXMWJYdxpAHd8xcChjmoA1xyxXZo2EnqaCNChtsYkhnHNkfkBn4RfNhGgs2PHjkEjAethUYGoQFRgvVQgGgfr5Z2WeaJZMOpGh6Dqly5d9QuXWCvP+VKjXOrORDmm207T5eOwWfsiL6ep+XLd8VyOrVgtdqu/47XmyDnpssfXxW7FliqOzm2lr6MBgMsQ8Gq1lfakhda81yLv6M6d6SvHj6eXP/nkguld+9RT6Yr89Am18u+EB0eKE4NfL65/X+MOyFQbsRhXcWKjcqnFhE9pqo0x6tc0Sr7To0bJJd6iTS70VYe4YuAQx3pNH2Ol1bg1XOMypmLQJ96jAT/VoQbwmg7GStPv/XFNBPg6bcXA4e9tS36t/LgvAioVFhWICqzXCizcE1mvVVgH8+YlCFyWX7AogX65siStPPAd12HU1qXuNCherjs9l7fDoDWNv/N1WJlzLS5wl2cr5mK3+jqe06vlCLw05+/iTIO5GMhjGk3nW85tNW1PcrPEgwcPxs0SV8ib/ED+d/gni8bBng0bxmanv8c8eFMMAj04ufDTvzsejAFXfeU7XDH4kl/DazHh60x1qK0a8DmR+8n3OMC406jhNe0ejZo2cNU/kTdGTjblYpR8xYlhvCdH1VBt1QDOmitODOOqo7lgrLRyf2BcI6Gm7XDND3HH5VjjUzvui1C+e7EdFYgKrOUKRONgLb+7eW5r4VIEvEX8kta3q/xCx5jDZu3r9DQvXXdcl2Mr5vRq855G0/nqvLju8nG+DuvJe5o4PbE5r9W+RPMAT05otbhZYmulVgdPf+f1wKgHVy5mTZ3yb5EHgconF36zwGsxoe+sFpM6Ok5/Ysyd2xgnhvUaTm1wXE2AU0c1aji5GHfawGvm+IrBryfHllx0TuQrhpjENRfmgfGaaSOhp4kAPerXYmqOzA9+PXxy45IGVC4sKhAVWMsViMbBGnx3eVYBl/xS06nqlyXxaXjQcP7U1qXuBCherjs9l7fDWn0Rs9XfaZY5Y9vxXAxwHe6wVk3n67BWPeTorNXfxW7FXAzk0urfynPzW+3Y3r17U++TFnhvhKW838GtZ5yRLnvwwZPKjf+6t/yX/STHAE6qQPl3wIMjxYnBuRfXv1MepKkGNKlf4vfd96r08MMpXzqX0imnpLRnT0qnnfZZuAxM+dTAgIs59Dj5Z03jZOYQcXzFwGIuNbyWn/KpAT2HK6YxVRs4a451Z8pXbg3XuMxRMcQg3qMBP9VxGuBojtguTfcfxjUR4OtyVAwcxnT51TSAj+LHJQ2oUFhUICqw1ioQjYM19I6uhbMLyi90vj36BT0Kc/7T+Do9xtdljdca2/GcpuM5DLk5vFVT58b1Vl8XtxVDrKWKw3nVluceOZJuNIOXbNpk0JUHsQHAhkBLhuTSt8VnWs4nnngibdi4cYFMeT3/gsHYmKoC+rfIgzfFIO5wYhhXvuL6t8uDMce/7bZXpS9/OaX770/p6afRLIBmSt//fkq7d/9kevGLU3rJS1LatOmvxuZSi4m4pTHv8uaIwJ0O+dDRefbgqgsd1qVHA341vupTG3xnysU4+YoTw3gtpsNVQ7Udd5S26mgu8ClNmwgYG9dIqGk7XPOGNt9/5QJnjuP4cUkDqhUWFYgKrPYKRONglb+DPKsAy/ILjVMrv9CAO24rz/kyVrksv9jLcW47TZePw2bti5ycJnPVpeO5HB0GHYdPozmNns6L6y6XWt4udivm4jjfaWNzXq3Lq7ZubaWuSB4bAGwItCQJ7r59+xLOWghb2xXQvzEeGGHGDlcMHPIVJ4bx8m9aD7BuvTWl224bNgxOP314pkHuH6UHHkjpnntSyveCzDmkdPnlfbloTMZDLr3mdHSe0ONcFSeG8Rq+1NrIpWYuF8XgxzrW5lPDVcdpQJv1Ug3FnQbGa6b7G7NsIiCe5si8gddydHxy45IGVC4sKhAVWI0ViMbBKnzX+EQENg34ZaRT0S8t4o6HsVZuzZ/6XOqXNzG3dHouF4dBbxp/5+uw1rzBc3m2Yi52q6/jOb1ajq1zdHGmwXpynCaOm996wtA86H3SAu6PcOjQofVUpnU/V/0b0wOjHly5KKjq8O/9kUdene6+O+VH2aX02temdNFFn5mv/b33Xpm+8pWUbrhhCO3b98wCDdVXbYczHsV58Mrtcuk0wFEd1XB8xeDLHBUnthjaGlPzBq65Y7s05SvX4Tof6HBONdxpwE/51KjhqgGO5ojt0nQ/pKeJAB1qa0xiGK/lPQk/LmlARcOiAlGB1VKBheeHrpY37mWJAABAAElEQVSs12meaBTcn8/tPJJPoT569Ohgh0a/qPBlxhdLhHG+iGFJnn4BkqeaDlMdruNLmi9itSU1dZz5jMPoqznSV+fiMOfrMM1B18lVzMVpxaiHpRr9J8Go6XyhO8roq/kwF/V1GHSJawyHMY7jjYtDPeXVYqv+el/HzRJ7refmir3awV/ZFdC/M/1ba8F1ZsonjvsZ7NiR0nOfm9IFF/wl4cHy7LM/k573vJS/31L63vdw0P6q+c8VatGB21w6nBiW/NzRzzcd13VqcsmxmgZ5WKo5XDHlz0IbsVVfc6npK4frysU6TXFiWNZiOlw1VNtxR2mrjubi1rlvos0ExyPmtBWr5Y050Fr45GJJPhoJ2Mfjvh7/UaTcWI8KRAWiAstVgTjjYLkq3xiXZxVwqV9YlNAvK2LT8KDh/Kmty54vYvXDusvbYS4Xx6tpOn+Hlflhu8Zz8Vsxp9nq63item5+tTm6ONNgLkfEnkbT+dbmGHjqvlliPGkhfmtYAf1bG/efYeXCn3ziu3YN71+AJaz8bNi2bYjjcgW8tmyZ/KwDxhwqnpwL8XFL1eF8yrz5H2nHhX4P3qINTebSow0/1WfewGvm+IrBjzqzyEU1oD1unpoL84Cfs3KfpedsBNWuxdTcmTfycHzlgkM+ueUlDVvyaTp8gR8WFYgKRAWWugLROFjqijfGQ6NgFs0ChCu/nIDxiwnrNIdxTJflF6+O6XpNz+XjMOfveA5zvg7TfLle47k4rZjTbPVFXo7bqsl56bLV18V12LQ5Os1WTOcV6/UKTHq/AyjSt64eI7OowK5jx1LavLK/lvXvkgc6mHsrvmXLp9Nll70m38ju09lnWDXVwQ0T8YSFZz0LTYPPDAiqDYB8xYlhfDx+4j/D4MP4magHh8ORhT9r2vQHmxrKBc4cW3By4ee0gasO+YqB43BiGFdtbDN3rDtTvnIdPk0uqq06zF0x5Elc8wCuOtguTfdnepoI0KG2xiSGcc2R+QGfhF82EaCzI5+6g0ZCWFQgKhAVWKoKrOw9lKWqwgqJw0YBlvrFounpFxFxx23lOV/qlkv9gi3HdNtpunwcNo0vcnD+DtN8ue54LkeHQcPh02hOo8c56dLlUsvbxW7FXBznO21snVust1WADYDemyVCnb5tkYI1aQVeN/w3+6TuS+pX/l3z4EhxYkiM+MaNf5G3NsznSvyGG16Tvv71YdPg4osXfp6POyCjBkUZV3Fi5Lilfn5pTMdVbYxTv6ahfHLh53DFlkobcWq5Y6y0GreG65w4f8WgT7xHA36qQw3gNR2Mlab7OOOaCPB12oqBw9+hlvxa+XFfBFQqLCoQFViOCkTjYDmqLjF5/RqbBuWXDqj6hUNXx+vh1vypz6V+kRJzS6fn8nYY9Kbxd74Oa80bPJdnK+Zit/o6HvJp1QS3tFZfF7sVczGQR6t/K6+cW2z3VwANgN6bJcaTFvrrvB499O+YB2+KoSYO37z5tenv/z6lb38bnxkpvfCFafBIRq2hfsbwYAzjqk/tGk4uz3SgPnCOAaNOLSb9yuU4DfCZu+NivAfX/GatDT3VZ97AnSkX4+QrTgzjs54nNPm+1bRrucC3tHLfZ1wjoabtcM0PcZk31nv45KKRgH1JGM5AwJkIXB+sxI+oQFQgKjCjCkTjYEaF7JVBo4BNA374q0b5xcKxVq7jOYy6uiy/MHWsXHeaLneHzdrX6ZX5cttxXY6tmNNDrFZ/x3Oajsc56bLH12m2Yq1xnF5PfXRusT59BXCzxAMHDnQJHTx4ME1yk8WuIHPkPRtO/Dd6Ev/wWf4K6N+8HhiV+G23vTbdeGPKN4RL6cwzU76MIaWXvSzlm+2euJSBs6FO+bnDA9JSm341nOPl0vFrMUtfbjsNjKmOyxsczrOmUcMn1W6J6bThVzPHVwx+k85fdagBPVcXxcBhbWsa4DjT/aKeJgK0mGMtpubI/ODXwyc3LmlA5cKiAlGBxarAhvyBdfLFfosVbZ3r8qwCLvlBr2Vxb0crDzqO6zCNyXX9YiTmlk7P5e0w54sYjusw5++w1rx7Yrfm43itmJuL83XzA9bq7zRbMRdj2jq62LU5Bj6bCuCsg94nJ1x99dVxycJsyr+uVXhw9I1v/F+DMw3uvDOl7duHTQM0DjZswKUMQyOX21zWcB6kkccl+TfdtCN99KOnp5tuys99TJsGw5dfvim9+c25a5Ht8ssfnT+4HADygxoCDVZrMUsetns1avwevJZfj8Ykubv513Kp4T05zkIDOdd03HzGNRFKn5q2w2tzd1zEGcfn2QhxX4TyXYntqEBUoLUC0ThordQUPDQKZtEsQAruAMsdyDnMTWGaZkEtn9YcHc9hbi4Oc/MD5rguTivWqrcY9XFzbM2ndX61vGcdx+Xj5rdc2J2nnZZuvO++dNXWrfMpXJuvP8d/wy/ZNDzgmB9YpSs4i6DnfgeYZjQPVumbvYLSfuKJ/enLX07pm9/EqdUpvehFKT3/+UjwU/NZuoMgh8GhhvMA62MfOys3Cnbk1845fZzW/YX84t/xqXk9Xx8xZ2960/25kZDv0jhnTt9hoDMmfccte3Rq3F7c5dirUeM77VE1cHyHQaMWs4Y7nRq3hjuNUfPpaSTUtGt4b46OT202ETCXaCSMekdjLCoQFdAKRONAqzHDdTYKuGw96JqG53xrU5qmYeAO+Bzm8nE8hyFv5+8wN0fHc3EcBj2HT6M5jZ6bH7BZ59OaI2I77jQYNFeasXFQ7ny9YOPGNdM4QM2jebDSfvPWfj6f//z+9K1vpYRHL150Ee5ncHgw6fJvjZXowUvuxz52dvrjPz6bUnPL0Y0DkHDmAZoHWKqV+hxzOA/SyBm3dBrwqek4vsOgUcOXQxv5OKvl4vDafGq400AONb7DaxpuLsBWehMBOWJObBzEUxpQkbCoQFRgVAXiHgejqtM5xnsWzKJZgNDuQMwdLDrMpT7rZkFPjm4uDnNzcZibX43n4rRiTrPV1/GQd6tm6xxdnGmwnhynjePmGNjSVoBPS+g58yBulri079FaivbQQ/sHT07ARZJXXpnSnj3DpgHmqJ8neuDWgyv35pu3maYBIm3ML5xloGccAD9hOEPhIx95Iv3O7zyz4OBS9cflWH7WjzvwrGmrjmo4vmKYDXNUnBjGZ62tMVUbuOaO7dKUr1yH63ygwznVcKcBP+VTo4arBjiaI7ZL032ucU2EmrbiGq+W9yT8eEpD+c7FdlQgKlCrQDQOapXpwNEoYNNAP7QpoR/wxLBs5Tqew1Sb6/rFRay2dJoud4fN2hc5Ok2Xu+O5HFuxWuxWf8drzXGa+cHXxW7FWnN0ej2x3RwDW94KTPKkBdwf4dChQ8ubeERfdRXAmQZ4egKu9rk/Xw1w1ln7B+tPPnmigYBJlZ8zPKhrwcm9+eZTstJ38uuM/OJlCnl10DB4MVZG2FfzGM422D6TXBBIP2P1IBBjpdXmWdNQPucPTYcrBg75qg2cOSqfXIz34qpPbeg4Uy7GyVecGMZ7clEN1e7RgJ/qaC4YK033xcY1EeDrtBUDhzE1b+B8j1r45MKP/HhKA6oRFhWICrgKROPAVaUB41kFXPIDV13LD3OMtfJqXOevMbmuX1LE3NLpubwd5nwRw3Ed5vwd1pp3T+zWfByvFavNxfm3ztH5ToP15DhNHDe/wFZWBSZ50gKaB0v1pIWVVa3IZpIKHDmyP+FGiPfeO2waPPBASrt340AHr/0LJC+9NKWf+zk8XWHYUNDPHz3YcTix4TKLp9ytSFjihTMNctB5w/Yj+fXwHIJ1vGCb5r/PxsUEm3GxTr5iiutnLw8AMV4z1aG2asCPOo6L8V5c9Z02NJnLLLShVzOXi2LwczkyP4zXclQdpwFf6vRowM9ZuX82rpHg8oNuDe/JUbnQ5DypXT6lAZc18AV+WFQgKrC+KhCNg873G42CWTQLELb8wAbGD2us0xzGMV2WX0Y6Vq47TZePw2bt6/TKfLntuC7HVqxVD/EXQ5Pz4rI1H5fLYuTo4jisFpvziuXqqACaAD1PWsCTGXCPBF7usDpmuXKzPLpzZ/rr/Ez21+WbcNI+mZ/PjhtyXrF59X9d79qV0nOfm9JLXpLSPfcMZzjJkzf1M4gHOlCr4cNIaAagcYDld4fQmJ+qp+twY1zFiWG8By8/93nwCh1nNW3VoYZyocUcW3By4ee0gasO+YqB43BiGFdtbDN3rDtTvnIdPk0uqq06zF0x5Elc8wCuOtguTffdepoI0KG2xiSGcc2R+QGfhF82EaAT90VAFcKiAuunAqt/T2QJ3is2CrjUD1yG1w9nYtPwnC91y6V+6ZRjuu00Xd4Oa/VFvFZ/p6n5ct3xXIxabMdt1XS+DmvV45zKZau/i92KuRg9NWuNU84ttldPBfbu3Tt4akLv/Q4ww2gezO59RrNgLdqmTYfTy16WBq+W+T399HhW+bnEgyPge/c+nl9bUu5vTWS//Mtbug6wRuXCBJgftpWvuH5W60EgNXSpGsCpU9NQPrnwc7hiS6WNOLXcMVZajVvDdU6cv2LQJ64awPleKJ9cjNdw1aEG+M50f25cEwH+TlsxcBizJb9WftwXAZUKiwqsvwpE42DEe74Wzi4ov0AwXf3y0Ok7vNW/1dfpaQ5cr/FcnFbMabb6Ii/HbdXkvHTZ6uvitmKIt1RxdG6xvnorwAbAcjYPHsgHffgve1hUoLcC+tm4b9/jCa+DBzfmV7vSvn0b0xvfuDk3HRb+DupnKQ/GoKoxxx1IKhe+5NfwWkz4OlMdaqsGfJi742K8B19MbeSi+swbuDPlYpx8xYlhvGee4Dsd1QCHNVecWE0DuDNtImB8XCPB5Qc/h2t+4IzLscandtwXAVUMiwqs/QrE4xiL9xjNglE3OgS9/AAFxg9PrNNaec6XGuWy/CIpx7ntNF0+Dpu1L3JymsxVl47ncmzFarFb/R2vNUedF9edL8ZcnGkwF8fpTRub81qLy/XyOMbaezfJYxpxqQPOWmi1W884I1324IML6PhvO07LX4+NA16q8ORjjy2oyVq5VGHBpJZ4AwdGN9/8g0HUgwdPyWchHJ3PAI0CNAmwxAumB1LzxBG4HpCS36vRy3cxGbtc1rRrGo7vMMSp4T3aNZ1e7XLe3K7lUsNdXIdBv0cDfKdT0wDf2bgmQunj9B0GP5cf8En48ahHVC4sKrB2KhCNg/xeslGwWJci4NfFHcg5zP1qLUWzoJajO9h0mJuLw9z8ajwXpxVzmq2+juf0MBfHbZ2j850G68lxmjhufmsV+8Tx4yftRL0+3xxqvdiBAwe6poqmQc/NEtE4uOPuu9OG/CxxtWgcRONAfx9mvX7w4J708Y/flWWHj2NE0+D9799y0t8649YOpGp47wGW03EY8qnhtZicgy57NWr8HryWX4/GrOZfy6WG9+Q4Cw3Ms6aj7yPXZ9FEqMWszb2Xz/mgkRD3ReA7F8uowOqrwLq+VGEtXIqAXzl3wNh6YDhrX6dX+7Nw3Na8Hc/pIbbjtmJO0/m6Obb61vQc7rBp4ji9Ws3cHNc6dtXWrWt9iiPnFzdLHFmeGFxjFSg/D3nQpDgxTL2G62cyD5hG8Z2OYvBlXMWJYbwWE2OlqQbGqKMawJm78snFeA++mNrIRfWZN3BnysU4+YoTw/hyzLOWC/Iprfzn0rhGQk3b4Tp3xNX3v4dPblzSUL57sR0VWF0VWHeNA55VwCU/zPRtKz8oMTYNr+avMblefgEQL5et+UwzF8Rs9Xf5lDlj2/FcDHAd7rBWTefrsFY95Ois1d/FbsVcDOTS6t/Kc/MLbP1UIG6WuH7e6/U+U34m6oERMdSGuGI1nFyMl5/VPCBVHeX34MrVXDQm42G8ZqqjuTgd5UKPfMWJYbyGL7U2cqmZy0Ux+LGOtfnUcNVxGtBmvVRDcaeB8ZrpfuQsmwiIpzkyb+C1HB2f3HhKAyoXFhVYXRVYN42DtXB2AT9sy18x/WDmmMOcv+M5zPk6jPF1WeO5OK2Y02z1RW6O26qpc+N6q6+L24oh1lLF4bxiGRVYCTdLjHchKrBUFdDPYz0w6sGVi7xVRz/Dew4kVUP1x+EaD7kwJtad1bRVRzUcXzHEYI6KE8P4rLU1pmoD19yxXZryletwnQ90OKca7jTgp3xq1HDVAEdzxHZpkzYRoENtjUkM47W8J+HHUxpQ0bCowMqvwJpuHPCsAiz1g0zfFv3gI+64rTznS91yqR/o5ZhuO02Xj8Om8UUOzt9hmi/XHc/l6DBoOHwazWn0OCddulxqebvYrZiL43ynja1zi/WogFZg0ubBvn37um6WqDFjPSqw3BUoP2d5UNeCk4s5KF9x/Wwfd0CmGtCkTgtOLvxqMTFWWk27pqF8jelwxRCXfNUGzroon1yM9+KqT23oOFMuxslXnBjGe3JRDdXu0YCf6mguGCtN9znHnYkAX6etGDiMqXkD53vUwicXfuTHJQ2oRlhUYOVVYM01DmZ5o0O8XeWHITB+sGGd5jCO6VI/uBUv152ey8Vh0JrG3/k6rMy5Fhe4y7MVc7FbfR2vlmeNC77arPNxcV0M5OC402A6r1iPCoyqAJoHt9xyy+A1iqdj733ve9OhQ4cUivURFdh17FhK+YkSpeFmkWHLXwH9rNWDHYcrhszJV5wYxsvPfHdApvyajsMV01w0JuNhvGaqw1xUA37UcVyM9+Kq77ShyVxmoQ29mrlcFIOfy5H5YbyWo+o4DfhSp0cDfs7KfdFxjQSXH3RreE+OyoUm50nt8pIG3GCRL/DDogJRgaWrwJrZG8FZBWwa8MNGy1h+MHGslet4DqOuLssPaB0r152my91hs/Z1emW+3HZcl2Mr5vQQq9Xf8Zym43FOumz1rek53GHTxHF6tZrp3GI9KtBaAdwssfdJC2ge9DxpoTWXtcp73VNPpbxXvFant2bmpZ+3PNDB5Hpw5cJXdfS7YNIDSdVXbYdrPOTCmFh35jTAUx1qKBcc5tKCk1vTBq465CsGjsOJYVzzxjZzx7oz5SvX4dPkotqqw9wVQ57ENQ/gqoPt0nQ/taeJAB1qa0xiGNccmR/wSfhlEwE68ZQGVCEsKrA0FVj4DKyliTmzKLxvwf3335+wjg8U/SBCIHxg6YcWMHD4wjaMPOWSU2oSH3rWf+KDWD+Ma0zqaRyXj8OcL+KQqzEdRn/lOUzHuU4elmouTitGTacHDRr1xmHU0xydL3XLJf0Vp38rNkmO0O6N4/LR2Doe61GBSSvQ2wTAWQoHDx6cNFz4RQVWfAX4Wc0lE+Z2+Tnci/N7SL/HEMPpKKZxFWd+NQ3gtZjqy3XV1pg1DeVTA0uHK7ZU2silljvGSqtxa7jOiVqK1eYJPZryiWFZw2u5qC/Xue/asv8KH6etWEvek/DZRDhy5EjCcQCW+AdiWFQgKrB4FViVZxywYYCy6AcSy6QfusQcD2Ot3Jo/9bns+aCljy5dPg5z+Tiew5yvwzQvrtd4Lk4r5jSdL3JwuMN6NDk3Llt9XVyH1fKedZxabM4rllGBWVRglk9auOzBB9MdJqk9GzYYNKCowOqogH4W8z+simEWPTi58NPvjZb/6mpc6ijWkkstJnydqT5jqgZ8mLvjYrwHX0xt5KL6zBu4M+VinHzFiWG8Z57gOx3VAIc1V5xYTQO4s3KftudsBJ1nT97Io4dPLhoJbBzgUgaciQDDelhUICowmwqsmsYBmgUwLvlBoWXQD0ni0/CcL3XLZfnhWo5z22m6vB02jS/iO3+HMVddOp7L0WHQcfg0mtPo6by47nKp5e1it2IujvOdNjbnFcvZV+Br+UyiSzZtmr3wKlKc9GaJmCJ9Od2rtm3jaiyjAmuuAvr5rgdvPbhyUSDqlN8nPFBTPrnwmwVeiwl9Z7WYquPyhhZzr2nU8Em1W2I6bTdvYo6vGHiTzl91qAE9VxfFwGFtaxrgONN93Z4mArSYYy2m5sj84NfDJ5dnIyAmGwdxSQOqGRYVmK4CK7pxwHsWzKJZgDLphxLLxg8ZbmPpMB3nun6AEnNLp+dyga/DW/1bfZ1ea961HF1sh7nYjteKIZ9WzdY5tsZu5bn8pq2ji+3mF9jkFbjztNPSjffdN9jBekHe+fj63GmiWA870QD4+Mc/3lwOcONJC83lCuIaq0D5uc2DI8WJYeq9uH7X8CBNNaBJ/Rac3FG5uJjgO9OYql3TcHzFEIM6NbxHG3qq06MNX9Yc685qudTwSXPRPJwGcnO45gGO6mC7NN0HHtdEgK/qU1sxcIi7/EqNVj6aCLDybAQ2FAaD8SMqEBVoqsCKbBygUcCmQfmhglnpB4rOspXreA5Tba7rByWx2tJputwdNmtfp7cYebu5II6L77itWKuem2OPb2s+jtcax/kib4c7zM0xsNlWgE2D2aqufjWcPdDTOMCM40kLq/99jxnMpgL6eT7uINVxkUUN1+8fHoyN4jsdxeDrciSG8VpMjJVW01YN+DB35WvMHnwxtZGr6jNv4M6Ui3HyFSeG8eWYZy0X5FNauW88rpFQ03a4zh1x9f3v4ZMblzSU715sRwXaK7Bi/nWGZgFei3WjQ5QEHxr84GCJHMYxXeJDsfxg1HGuU0/j4EOPL/K4rR+Izhd8culbw+ivPIfpONfJw1KtNbbjOU3ysKS1Yq161C2X9FecsR02qxyhPSrOJLHVJ9ajAstZgd6bJSLXuFnicr5jEXslVoDfEfq9gzwdrpjyFdc58ruPS47V+D24crFOYywsW0x1lO90lKsxFVeNGr7U2ppTue5yUQzrtNp8arjqOA340VRDcadBH7fkPvOk+83QrMXUHDV2D59cNBEeeuihwY0VcXPFuMGiVjTWowInV2DZzzhgwwCp4Q+5NP3g4pjjYayVW/OnPpctH3jkOk2Xj8Nm7ev0mGe5dFyXYyvWqoc8FkNzsedXy7t13q1zdrxybrEdFVgJFZjlzRJXwnwih6jAcldAP//1v6s9uHIxH9XR7yv+V7vGV1w1enCNh1wYE+vOatqqoxqOrxhiMHfFiWF81toaU7WBa+7YLk35ynW4zgc6nFMNdxrwUz41arhqgKM5Yrs03ZfuORMBOtTWmMQwXst7En5c0oCKhkUFRldgWRoHaBbAuNQ/cKarHwbEpuE5X+qWS/2QK8d022m6vB02jS9ycP4O03y57nguR4dBw+HTaE6jxzmVy1nn05pjrT7OvxUr5xbbUYGVVgHe8LDnsgVy6bvS5rQc+RzduTPtOnZsOUJHzBVagfJ7ggd1LTi5mJryFdfvynEHZKoBTeq04OTCrxYTY6XVtGsayteYDlcMcclXbeCsi/LJxXgvrvrUho4z5WKcfMWJYbwnF9VQ7R4N+KmO5oKx0nT/elwTAb5OWzFwGFPzBs73qIVPLvzIj0saUI2wqMDCCixp42AtnF3ADxQtY/lhxTGHt/q3+jo9xtdljefitGJOs9XX8ZBvq6bOjeutvi52K9aTY6um43FOsYwKrJYKsAHAhkBL3uDGzRIXVurPN29OTz722AIQj6e8IuNhUQH9vtCDHYcrhsqRrzgxjJffoe6ATPk1HYcrprloTMbDeM1Uh7moBvyo47gY78VV32lDk7nMQht6NXO5KAY/lyPzw3gtR9VxGvClTo8G/JxpEwHj4xoJLj/41fCeHJULTc6T2mgi4IW64KaK+gI/LCqwXiqw6HsiaBas1Bsd4k0uP7hqbzw/PHS8/KDBmMNm7Ys4ThN4aY7ncmzFoD9rzVa9cm49uYDbOkfHa83R+fbEBjcsKrBaK4DmwS233DJ4tc4hbpbYWqngRQUWVkC/b3igA0YPrlz4qo5+7016IKn6qu1wjYdcGBPrzpwGeKpDDeWCw1xacHJr2sBVh3zFwHE4MYxr3thm7lh3pnzlOnyaXFRbdZi7YsiTuOYBXHWwXZruj/c0EaBDbY1JDOOaI/MDPgm/bCJAJx71iCqErYcKLErjgJcgcKl/mCyq/hETa+WB77gOo7Yu9cNJ8XLd6bm8HeZ8oe+4DnP+DitzxnaN5+K0Yk6z1dfxnB5yd1zgpTl/5zsN5mIgj2k0nW85t9iOCqzmCuBmiQcOHOiaApoHepPFB/LNuvBf9rCoQFSgrQLldwsPjhQnBsVeXL8Pxx2QqTZiMa7ixEblUosJn9JUG2PUr2kon1z4OVyxpdJGnFruGCutxq3hOifOXzHoE1cN4Hz/lU8uxmu46lADfGe6nz6uiQB/p60YOIzZkl8rH00EWDzqcVCG+LEOKjDTxsFauBQB73n5YQNMP2iwDXPYrH2d3jD6yT8d1+XYirXq1Wrh4vRoljNs9XVxFyNHF8dhtdjl/GI7KrBWKoAmAJoB4+yNf/RH6ZF7701/+pu/OXjSwulveMPAJU7LH1e5GI8KjK6AfhfxoE4xePfg5MJPv4t5MAZc9ZXvcMXgS34Nr8WErzPVobZqwIe5Oy7Ge/DF1EYuqs+8gTtTLsbJV5wYxnvmCb7TUQ1wWHPFidU0gDvTJgLGxzUSXH7wc7jmB864HGt8aqORUDYRoItLG8KiAmuhAlM3DnhWAZf849HilH9oGJuGV/PXmFwvP3CIl8vWfKaZC2K2+rt8ypyx7XguBrgOd1irpvN1WKsecnTW6u9it2IuBnJp9W/lufkFtjIrcG5+NNONJrVLNm0yaEBagZ4nLTz3xS9Ov/pnf6busR4ViArMsAL6/aQHRj24cpEadcrvTh6QKp9c+M0Cr8WEvrNaTNVxeUOLudc0avik2i0xnbabNzHHVwy8SeevOtSAnquLYuCwtjUNcJzpfn1PEwFazLEWU3NkfvDr4ZMblzSgcmFrrQITNw7WwtkF/OPWN1U/NMbhrf5O0/k6THPgeo3n4rRiTrPVF3k5bqsm56XLVl8XtxVDvKWKo3OL9dVTgau2bl09ya6wTCe5WeIKm0KkExVYcxUovx95cKQ4MUy+F9fvVB6kqQY0qd+CkzsqFxcTfGcaU7VrGo6vGGJQp4b3aENPdXq04cuaY91ZLZcaPmkumofTQG4O1zzAUR1sl9bTRICv6lNbMXCIu/xKjVZ+XNKASoWthQp0NQ54VgGW5R8ai6F/aMQct5XnfKlbLvUDpBzTbafp8nHYrH2Rl9PUfLnueC7HVqwWu9Xf8Vpz5Jx06Xwx7uJMg7k4Tm/a2Dq3WI8KrMcKTNo8uPnmm9OrL798PZYs5hwVWNIK6HffuINUx0WyNVy/a3kwNorvdBSDr8uRGMZrMTFWWk1bNeDD3JWvMXvwxdRGrqrPvIE7Uy7GyVecGMaXY561XJBPaeUxQM/ZCDpPF1Pnjrj6/vfwyY1LGsp3L7ZXSwXGNg74RAQ2DfhLrxMs/6Aw1sqrcZ2/xuR6+UFBvFw6PZe3w6A1jb/zdViZcy0ucJdnK+Zit/o6ntOr5Qi8NOfv4kyDuRjIYxpN51vOLbajAuu9Amge9DyiEfUCPxoH6/03J+a/1BXQ7zQ9MHK4YsiTfMWJYbz8DuaBWo3fgytXc9GYjIfxmqlOLXfqKBd65CtODOM13OWo3FlrQ69mLhfF4Ofmv5jzREzqay7MA+M10+ODWTYREE/fI+YHvJaj45MblzSgcmGrpQLVxsFauBQBbwL/MPUN0T9g4g6bta/TY/xy6bgux1bM6SFmq7/jOU3HK+eG7Vbfmp7DHTZNHKeH3Gs4xsKiAlGBkyvQerNE9Tx48GDiGQuKx3pUICqw+BXQ7zk9MOrBlYuMVUe/m3kQWOMrrho9uMZDLoyJdWc1bdVRDcdXDDGYu+LEMD5rbY2p2sA1d2yXpnzlOlznAx3OqYY7Dfgpnxo1XDXA0RyxXdqkTQToULsWs5a38qkBvVH8uKQBFQpbyRVY0DjgWQVc6i89J6G/8MSm4UHD+VNbl/qHr3i57vRc3g5r9UXMVn+nWeaMbcdzMcB1uMNaNZ2vw1r1kKOzVn8XuxVzMZBLq38rz80vsKhAVODkCvTcLJHePEthPTUPdh07ll6eH0V5LYswt4ynTBQFic0lrUD5nciDuhacXCSsfMX1O3vcAZZqQJM6LTi58KvFxFhpNe2ahvI1psMVQ1zyVRs466J8cjHei6s+taHjTLkYJ19xYhjvyUU1VLtHA36qo7lgrDQ9lhh3JgJ8a9oO17zhy/dIucCZo/LJxTj5cUkDqhG2kiowaByshbML+EdWFlf/KDnmMOfveA5zvg5jfF3WeC5OK+Y0W32Rm+O2aurcuN7q6+I6bNocnWYrxjnFMioQFeirABsAbAi0eJNL3xaf1c7Zs2FDen08umu1v41rOn/9vtSDHYcrhqKQrzgxjJf7C+MOsGo6DldMc9GYjIfxmqkOc1cN+FHHcTHei6u+04Ymc5mFNvRq5nJRDH4uR+aH8VqOquM04EudHg34OdMmAsbHNRJcfvCr4T05KheanCe1y0sa8IhHvsAPiwosRQU233///Qt+4TVo+UuMMf4CT8Jzvqqj6+Ufs47putN0eTtsGl/k4Pwdpvly3fFcjg6DhsOn0ZxGj3PSpcullreL3Yq5OM532tg6t1iPCkQFJqsAGwBsCLSogLtv376EsxbCogJRgZVVAf2+5YEOMuzBlQtf1dHv+EkPJFVftR2u8ZALY2LdmdMAT3WooVxwmEsLTm5NG7jqkK8YOA4nhnHNG9vMHevOlK9ch0+Ti2qrDnNXDHkS1zyAqw62S9Njj54mAnSorTGJYVxzZH7AJ+GXTQTo7NixY9BIwHpYVGCxKrBRf2ERBL/YfDEoOHwRU65ijucw9eE6/mD5IuaW1MOSxpz1D9Nh4NOfvliSOw6jr8Z2mOpwvcZrjU0eljSn6XitGHSpyRhY0l8xtz7KV/Om3iQYY2CpRk2HTRJHdWI9KhAVmE0F0DxgE+CPf+VXrOhdt966AH/ve9+7YDs2ogJRgZVXAX4Hc8kMua3fwxjrxcd996u+atdw5jcql1pM9eV6LWZNQ/nUqOWiXJ3PYmojl5q+5sv1GreG65yooVhtntCjKZ8YljW8lov6cp3HI9pM4JhbOm3FWvKehM8mwpEjRxL+GYwlbmwfFhVYjApsuOeee/Lf14kDUQ2iv+TEHdfxHEYNXbb+QcLHabp8HDZrX6en89J1x3U5tmJOD/Fa/R3PaTqezovrPb5OsxVrjeP0eurDecVyfVfgztNOS+fmL+DSvpYbnJds2lTCsd1ZgQMHDnR5oNmAmyyGRQWmrcDBg3vykzvuyjLDv+O9ezfk360Ft3yaNkT4FxXQ/7DqUA9e4+p/dSfVhl9N3+G1mBpf150Gxp1OjduLL7W2zrdcd7k4DH6zmGdNp6Zdy6WcB7fHnY1AHpY17Rrem6PjUxuXMuBMBBjWw6IC01Zgc3mQ1Xpw5nhIpoaXibY2DJxemTO0Fbv4rrvS3z/vedVclMu8HOZiO4wauqzxXJxWzGm2+jpeq57OS9db/V3sVszFQA6t/q08nVesRwVYgU8cP37STswLNm7kcCynqEDvkxZuueWWFE9amKLg4RoVWMYK6HexHuj04MrFVKhT7ifwoKnGV5wa0OvBazGh46ymrTqzzntSbeTPuvTk7eZNzOWiGHiTzl91qAE9l7ti4HCeNQ1wnOkxzLgmgmpDizkqTgzjmiPzA97DJ5dnI0CfjYO4pAHVDJu0AoMWO3/BShH95eWY4zqMfF3qH5ri5XpNz+Wj2MU335zyOTopPf54uvjLX05pz570tcsvH8grj/Ec5mI7jBrl0nFdnFasVQ95LIbmYs+vlnfrvFvn7Hjl3GI7KhAVWLoKxJMWlq7WESkqsJIqUH4f8+BIcWLIuxfX/YdxB2SqjViMqzixUbnUYsKntJp2TcPxFYM+c6zhPdrQU50ebfhqzbFdWi2XGj5pLpqH00BeDtc8wFEdbJemxzbjmgjwVX1qKwYOcZdfqdHKRxMBxssYeDYCGwqDwfgRFRhTgc3lL6v+ktK35AB3GPnlUv+oyjHddpounxK7+K/+KqUvfnHYNDjjjJSeeiqlO+9MaefOdMm3vpXynbXSVy+4YBCq9GV8F9th5OvS8Vwch0HH4dNoTqOn89L1WefTmmOtPs6/FdN5xXpUICqwfBWY9GaJyJi+y5d9RI4KRAVmUQH97h53kOq4yKGG674LD8ZG8Z2OYvB1ORLDeC0mxkqraasGfJi78jVmD76Y2shV9Zk3cGfKxTj5ihPD+HLMs5YL8imtPN4Z10ioaTtc5464+v738MlFI6FsIkA3GgmoQlitAvMX9ZW/kHDgL5c6O0zHuV7+8RAvl07P5QI/h1/8l3+ZEs4u+P73U9q6NaVnPQuJp/TQQymhu3bHHSmdemp65vzzy9BTzc/lXcvR5e0wp+l4rRjyadU8qTgdvq35OJ7LD7k47jSYm19gUYGowPJXgA2AeNLC8r8XkUFUYLkroN/zemDkcMWQN/mKE8N4ub/BA9IavwdXruaiMRkP4zVTnVru1FEu9MhXnBjGa7jLUbmz1oZezVwuisHPzX8x54mY1NdcmAfGa6bHQrNsIiCevkfMD3gtR8cnNy5pQOXCWirQdI8D/mKNE9Q/kHFcp6m/1PR32ALf668fnmnwqlel9KIXpa+ee+7A9YVoGOD11a+mfJtRyi34g5oH88oCTR0o1h3P5diKQX7Wmq16xdQGmz2+rXN0vNY4zheJOtxhbo6BRQWiAiurAmge4B4GeLUanrRw6NChVvqq4B3NZ8ntOnZsVeQaSUYFFrsC+p2uB0Y9uHKRr+rofggPAmt8xVWjB9d4yIUxse6spq06quH4iiEGc1ecGMZnra0xVRu45o7t0pSvXIfrfKDDOdVwpwE/5VOjhqsGOJojtkvTY6SeJgJ0qF2LWctb+dSA3ih+XNKACoXVKjB/xoH+cpHsMI7pUv8YFC/XnZ7+8pLvMOd7CR7Zdffd+IRI6bzz5psG8P/KhRemlF+X5jugp1NOWfBhyDhOk2O6rPFcnq2Y02z1dbxWPZ2Xrrf6u9itmIuBHFr9W3k6r1iPCkQFVkcF0DzofeziWrxZ4p9v3pyefOyxBW/anvwdd0XGw6IC67UC5fc/D+pacHJRO+Urrvsn4w6wVAOa1GnByYVfLSbGSqtp1zSUrzEdrhjikq/awFkX5ZOL8V5c9akNHWfKxTj5ihPDeE8uqqHaPRrwUx3NBWOl6XHTuCYCfGvaDte84cv3SLnAmaPyycU4+XFJA6oRxgps3vL+93N9sMQvUD4Mn7dv5rVdP/ETg+2dP/7j87j+0s+DlRX+8umw/qISd9hIX1yegJ2pbdsGy9Ifvn/36ldTfrB0egsIsuG4ZQzQW7FWvcXSlKkNVlvzaZ1fLe9Zx3H5lHOL7ahAVGB1VSBulri63q/INiqwnBXQ/QA92HG4YsiZfMWJYbzcZxl3gFXTcbhimovGZDyM10x1mLtqwI86jovxXlz1nTY0mcsstKFXM5eLYvBzOTI/jNdyVB2nAV/q9GjAz1l5PDWukeDyg24N78lRudDkPKldXtKA+yHwBX7Y2q/A5s3/+T8vaBSwaYDlt/LrXrx+53fmK7Hjx35ssH7hn/zJPOZW+EumY+UvJMYc1uo7eIJCbnQM7mVw++3pUtzT4JFH8GmRUn4G+99dfPF8eKc5PygrjudyhIvDHdaq6Xwd1qon01qw2urvYrdiLkZPzVrjLJhYbEQFogKrugK838H/l5+QcxfOKGsw3huBvg0uQYkKRAXWUAV0f4EHOpheD65c+KqO7s9MeiCp+qrtcI2HXBgT686cBniqQw3lgsNcWnBya9rAVYd8xcBxODGMa97YZu5Yd6Z85Tp8mlxUW3WYu2LIk7jmAVx1sF2aNhJ6mgjQobbGJIZxzZH5AZ+EXzYRoBOPekQV1rZtzrcRnDdtGnwno3iV9ijuKZDt2HXXJT0DAZj+4mEbpr+kQ2T40+Gt/vC9NDcK0sMPp5Sfs54eeCClP//zQbMAT1JIuDwhn43wYjxh4Ud+JN36D/+hhj5p3cUFyeXYijnNVl/HQz6tmuCW1urrYrdiiLlUccr5xXZUICqw+ipwa/6MviNfbrYBzd45O/0Nb0h782d8a+MAbmge7MtPz8FZC2FRgajA+q1Aub/CgyPFiaFKvbju44w7IFNtxGJcxYmNyqUWEz6lqTbGqF/TUD658HO4YkuljTi13DFWWo1bw3VOnL9i0CeuGsD5/iufXIzXcNWhBvjOepoI8HfaioHDmC35tfLjvgio1PqwjfkQO6F5gBfXcaaBaxpkeN7YQACAX8ryFxO/kPpLCR4xxemr/o53Evboo8OzC7B88kn8ZQ/uZYAzDdKuXUgqpfvuS+lLX0qXfe5zCH+SMbYOnBQnD7Zi0Bml6eKMw6g3rj6qw/UeX86Rvli2Yq1xqIelGvFxmI7HelRgsStw7pEj6So8qUUM15lfsmmTILG6GBXA2QO9TYDe+yMsRt6hGRWICqysCnD/Qvc7FOvFdXZu3wfjqq98hyvWkkstpsbRddUnrhpYpzkuxnrwxdRGLqqP7VGmXJ2n4urfM0/4OR3VwDpNcWJYOg0d13U0EfSlY269pu1wzU/zhm4Pn1w0Eo7kfaj7883pscRjH/noR5drYKunAiedcfDdnDte4+y+3/3dtOdd71pAK3/ZMOgw4PjlKs1xHTbwffzxlI4eHZ51gGbBP/7H6dYrr5yXvOwrX0np619P6aabUsIpr3P3aXBx4eTitGJOs9XX8ZxeLcf5CcuK83dxpsFcjFqO08SRacVqVGBZKvD6fA1f2NJX4JprrkkHDhzoCozmAfzCogJRgahAWQHdF2n5z7DjKwZ96pT7RO6/uspXHWpgvAevxYSOs5q26sw670m1kT/r0pO3mzcxl4ti4E06f9WhBvRc7oqBw3nWNMBx1nM2gmpDizkqTgzjmiPzA97DJzcuaUDl1o5tzHcFSE/lF842uDO/WpoGmTYwnnWAXzD9JcOgw/BLxNdQ4QRP/emrGP34izg4wwD/+cPNEXFJwotfTMnB8tZLL03ph384X1ORH211550nxQXJxWnFTsqnoqdxsE5jHG5jSU3FyMNylNF3vj6Z7HwdBl3iGsNhjON44NPoOw4Dn1z6xjIqEBWICqACvU0APM4RT1oIiwpEBaICoyrA/Q4uyeU2lmq9OPeVdJ8Mek5HMY2reEsutZjqy/Wadk3D8RXDOq2G92hDS3V6tMua01eXtVxq+KS5aEyngXGHax4t8+k5EwExVR/bMMU0pstvUj6bCOXZCIME4seqqMBGNA3yif6D5sH3OlPGfQ7wC0Vzv1z6iziKhzH6k4cl/Uss34EjpdNPT+nss1M688x061lnKWW4jjMRcL+DfGbCvu8ML75gDM0bZOIq4jCXj+MRw5LmMOphSXM8jpVL+itO/1Zskhyh3RvH5aOxdTzWowJRgagAn7TQUwnc7yCaBz0VC25UICrA/RndJ1GshmvllK8499N0Pw/jNb7DFcM6TXFiWNZiKofrqqHaqqG5K58aWPbgi6mNXFQf26NMuTpPxdV/OeZZy0Xz4ro2EfSsBI6Xy5q2w3XuWFfr4ZMblzRoBVfH+mY0DnBbqnzif7c98MEPpjP/1b8afFiUzvilKK38JcO4w5zvSRjOMnjWs1LCNcgSawHvqTw7nJGA+x/ksxPKWOV2Tz6tvtNqwr+0BXOcG2zNp5UH2aWKU84vtqMCUYGoACvApyXw6QnERy3Jpe8oboxFBaICUQGtgO4n6WnaDlcMGuQrTgzj5X4VTw+v8Xtw5WouGpPxMF4z1anlTh3lQo98xYlhvIa7HJU7a23o1czlohj83PwXc56ISX3NhXlgvGbaPOh5SoNq12Lqe8T8kEcPn1yejYC4eMQjLJ7SMCjDivox3zjApQqTmP7S8M1XHR0n7jDnC34NvzWfZXDZnj3DSxFyg+Cyb3wj3XzhhQwxXN5zT0rbtw+etnDTc54zwFxsh4HsYjtuK9aqN0x+4U/nC0Zr7Faei+N8p429cHaxFRWICkQF6hVgA4ANgTrzxAi4q+1JC7vypXWvy1P45IlppLjHhhQjVqMCS1wB3f/RA6MeXLlIX3V0n4sHajW+4qrRg2s85MKYWHdW01Yd1XB8xRCDuStODOOz1taYqg1cc8d2acpXrsN1PtDhnGq404Cf8qlRw1UDHM0R26VN2kSADrVrMWt5K58a0BvFRxMBxhsqopEQTYRBSZb9x3zjID/Q0NrzM5rvJJBwGcOD+ZX/f58ey5cAPLVtW9rxYz+Wtxb+kQ+A/EN/IUZh+gtFnsM4tmCJZsCzn53yLTtTyjdD3JdPm7n5oovSPjQMciMB9zbAZQzpggumyqd1Lo5Xm4vjLpjb3Ibzd77TYC4Gwk+j6Xzd/AKLCkQFogItFUDzAPcwwKvVcLPEQ4cOtdJXDC+aBSvmrYhEogLzFSj3a3hQ14KTCzHlK677YuMOsFQDmtRpwcmFXy0mxkqradc0lK8xHa4Y4pKv2sBZF+WTi/FeXPWpDR1nysU4+YoTw3hPLqqh2j0a8FMdzQVjpfU0EeBb03a45g1fvkfKBc4clU8uxslHI6FsImCcZyZgPWxpKrDpLSm9D6EeMvHyQw1T/n/9oHGQLwpIz8N2bhg8O/+H/7H8n/xn/9t/m7biHgNzpm/8KAy/COAq32HUKJfk3rN7dzoHlyngkYx/93cpXX99OucLX0jpxhtT+vznh09dwE0SX/rS9D08ojGbxqQu9cqxcrvm73jUZIyar45znb6qq+vkOawWx3EZh3o9vk6v5q/6sR4ViApEBViBe/P3yPfzPWh0RwFjl1ceeXnOOeekT33qU3RvWqLRsH///iZukNZHBW65ZUduQD2SJ4sLNfH/hw35d2S4PgDiR1SgswLlZxjde3DlYh9LXzo2iTZ8nIbi4+IxrltSWzWwTlx9HKZ5KLfEVb9Hp4Xboq25Ob5iWHdxHbYY89RcajE5H+VinQf0HC+XylftGq7+k/KpjYbH4/nJesePHx80E3AsAxt3GYbmEOuTV2DTr+TGAU4Icfc4OJ7xfEieMI4X7oewJb9hW/Pr2fmUyrP/5/9Mm/72b9MPnvvc9PTz0FY4YXiDS3MHqg4r/bBNnupi/Xv5bIJz8i9Peii3PtBAwP0MYLi3wXnnpfTyl6cvnnvuyIbB0GH4U/WJt2K1HKkzbkl/5bXGbuUxRskvt5HDNJjOIdajAlGBqICrwLMfeyx9LX+f6I7EqP+2PxtnmGXrOevgHpyBlg2XLYRFBVCBaBzE78FiVkA/zzTONDj2x/jq0enhIlfyGYtL4jqf2rpy6Y+l4urbgyt3qbRH5c551HKp4fTDUudEXLFJNVRbNbCu+oypS+X3NBFUWzVq8RSfhI8mAhoIbCLgzAQ0EKKJoO/mbNcHlyqUTQNcigB7PC83567O1ocfHvTlt+d1nIGwbe6FpsKW3DjYnF/AcJD++DvfObik4dQf/VEg86eZDDbmfuDgtcVqPPxyqd30ylemZ17xivTS7+aHSaKJgObBzp3pRtwDASZ8p1nqDV0WxhjoDKROxls1qaHLVt+lyrE1juPpvGI9KrDaK3Bn/vw7F5dBFYYD3Usq/xEvqLHZUIGr8llsPTbp/Q4Qg7498YIbFYgKRAV6KlDuH/HgSHFi0O3Fdb9RD+qcjmKIxbiKExuVSy0mfEpTbYxRv6ahfHLh53DFlkobcWq5Y6y0GreG65w4f8WgT1w1gPP9Vz65GK/hqkMN8J3N4pIGjYcYjNmSXys/7ovg3r3ZY5sfyzttT83dhAINAzQLYM/M/ef+yXwqKV6wx/JB+eb8HyI0E9BEOCVjbCJgufWGG9KWN70pPSevP5ObCDjEPv6Od6TjeR1W/uIMQPPD8bZkbZxL8MSP/Mi8h/7CAfxiPvMBVuLAnKbjtWKteohdmvMFpzV2K8/Fcb7Txi7nF9tRgbVSgU/kzzz9Esa8XpDv+Bu2vBVgA6D3ZonImr7LO4OIHhWICqyXCuh+F79PFEMdenBy4af7eTwYA676yne4YvAlv4bXYsLXmepQWzXgw9wdF+M9+GJqIxfVZ97AnSkX4+QrTgzjPfME3+moBjisueLEahrAnWkTAePj/rPv8oOfwzU/cMblWONTO+6LgCrO3jY/mE/93JCbAI+fgjZA/qXlqf4S65m5HeUnMwcv2JaLL05333prOiVfIrAt++eHIs6fjQDGoImQl9ve/OaE+yM8ng/4j7397fNNhAwtML7RC8C8gV+M09/ylrQt37MAjQPcnPH2f/APBg2LJ9761rRtrpFQ/gJBx2k6XivWowluabPOx+XtYiAPx50GK+cW21GBqEBUYCkrgAZA780S0WiIxsFSvksRKyoQFdAK6H6XHhj14MqFNnXK/T8ekNb4ilMDej14LSZ0nNW0VWfWeU+qjfxZl5683byJuVwUA2/S+asONaDnclcMHM6zpgGOM20k9DQRoMUcazE1R+YHvx4+uWgi4IWYvKFiPKUB1ey3Df/3+ec/83RuFpSPY2SzYIGkNBVe+B/+Qzr1ssvSkZtvTnf9yZ+kC3/2Z9ODf/AH6TjOOshOekkDzkYAhoN+vH6QD/YfzU0EPMlhZ24ClKa/LKfmpsHWuaYBGge35dfRwuE5uCHinPGXhNtcqmYv5jSdHnV12ePrNFux1jhOD/k63GE6t1iPCqzlCuBShRvvu2/+C5VzxRkHcakCq7H8ywMHDnQlsXfv3nTNNdd0+QR5bVXg4ME96eMfvytPCs+MSmnv3g35dwJ7J2FRgeWrgB4cMQuHYawX50Eadbns0enhQr8Wk7F1WdOuadT4PfhiavfOv5ZLDV+OedZy0feR6+OaCORxWdN2eG3u0OrhkxuPeuS70Lbc9LOnn/4+nHjLuw4MGgbSIBjIYFswNAye84Y3DA42t+UzFs58zWvS5nxzxJ1XXZVOyw2BLTi7IN9v4Il86cAjWGYRvJ7ML9xgMWVs2yc+kU7PL5xJsDkvN2TuU3i8otiO3DTYnMdxyQNet+fXMRnX1S0ve9lJB7/uwLcVw4E4uMrXdY3t1umvY85/GowxSo1yGzlMg+kcYj0qsF4q8HA+u+rufMPV8ktqT/4sPDMuV1gxvwa46WHPkxbiZokr5q1btkTi5ojLVvoI3FiB8nuHbg53GPiKYx+QL8WpW/InxVWb8bBUXLVr6+SrRk2H3FJrHL6Y2shF9Wu5MGfl6jwVr2mMw1VDtRkby3Ea4KhOjQ8eTLlY50H6cPTkn8pX7RquCpPyqc0bLD6a9/fwyEc0PXB81dv80JzW8vqmn8uNAzYLFlymwGaBNAzQXHjOL/5iuujd756viTt4RRNh18/8zOCFJsKWvP5wbhbghRsq4mkNeLGJsHmukbAtnzmw65OfTE/nBsLmf/fv0qa8zabBNzMfvjXb/tM/PRjCL4IzhzuM81ENx9NxrtNX+bpOXisGvuMyDvVqPOfbiql2rEcF1mMFonGwOt71SZ60wKcyrLQnLRzNN/XdmndcSrs2PwL5vGhWlWWZeDsaBxOXLhyXoQJ6YKThe3DlYj9QXzpGfYdhbFJc42G9psP4ulSu6ihe4xNv4S63NnPF0uWiGNbdnBwGPcVVR3HwaOPwFg1qYan8ld5E0Ec94vIGNBCiiXDi3dx8Tr5O9O7//t8HB+j5N+vEiKzxbITn5qbB8/ILhoPXFts6dynC2f/tvw3oT+QzCL7/h3+YHspLvaRhcF+EOWxHXj6MGPmFsyHuzq/H8qtmx9FgMA2DVszNxfnW4rf6O81WzMVAPq3+rbzaHAOPCkQFogIrtQK8b8FauFnin+dHCT+Zb0KshrNcwqICUYH1WQHdf9MDuh5cuaii6uj+JQ/qanzFVaMXdzFr725Nu6bh+Irp/BXX+cxaW2OqNnDWHOvOlK9ch+t8oMU5ccrpFQAAQABJREFU1XCnAT/lU6OGqwY4miO2S5v1fRGgz5i1vDVHcuE3io+mAQxnIcDikoZBGdKm3/zwh9+3M1968MS996bj+UXDQTv/2w8MTYPn/JN/MiiyFpp8XeINAqfkYXtTviQBlzTsftvb0uZ8NsLdn/50emTLlvREfmPwFuGFsxEQn68H83qLoUlRxoSfw4AzT6zTalyOc0lf5es6edNgLgZ0p9F0vsw1llGBqMCJCsQZBydqsRrWePYAzyZoyRlc+PGshRafxeQc37o1fTs/7vMH+QwDtR25cRBnHGhFpluPMw6mq194r5wK6EGdZuVwh8FHcewj8qX4OO1SZxxftRkPS8VVo7ZOfosGuaXWOFy1azmO0xgVU/VrOvRXruaieE1jHK4aqs3YWI7TAEd1anzwYMrVA/rh6Mk/la/aNVwVyFcu1ok7LjDy45KGYYU2oyC7c+Ngd77ZIW90iKFHvvSlISP/xP0M0DQYZzjQLQ36pRHDgf5Z/+W/pKNf/nJ69MYb071f+UramW9Gdmq+zuTU7ISzDcqTNndmDA9dxP9fvpdfODMB9sSOHWlHEYtxhozhz9Yc1YfrPb4udivWGsfpIVeHO4zzimVUICoQFViOCtx6xhnpsgdPbg0/kD/LJ/kv+yRPWnjve9+bDh06tBzTj5hRgahAVGCqCui+nR4AOVwxBCVfcWIYL/dFeWBX4/fgytVcNCbjYbxmqsPcVQN+1HFcjPfiqu+0oclcZqENvZq5XBSDn8uR+WG8lqPqOA34UqdHA37O9EwEjI+7PMDlB78a3pOjcqHJeVK7fEoDzkbgC/y1bIPbCLMQuOnhC/MLBgzNA7yemxsHNaNvOV4WHeMO23XppQmv9PM/n47mxgHs2L/8l+l4bh7oTRsHA/nHrvxCKwKJn59faCTcm68LPZKXMBejJ8ehysKfzt/FmQZzMWrzmSbOwpnFVlQgKtBbgXiiQm/FRvM/kU8H3FBcv39FPl1/UkPzAM2AHgM/nrTQU7HgRgWiAiutArpvyAMd5NiDKxe+qqP7qZMeSKq+ajtc4yEXxsS6M6cBnupQQ7ngMJcWnNyaNnDVIV8xcBxODOOaN7aZO9adKV+5Dp8mF9VWHeauGPIkrnkAVx1sl6aNhJ4mAnSorTGJYVxzZH7AJ+GXTQTo1B71+Jk7PpOuvOhKUFatbXr729/+PmaPgqGYLOi2s89Op77kJRxesCRXQfpNgkFvy5lnDl7H82MdeZnC41lMz1ng5QxY4mwELHEzqdOuvz6dlu+jsCOfufCDfHPFh+++O23KT3wocyq3s7s1zk/5uk6nVgx8x2Uc6tV4ztdhNX/Vj/WoQFRgfAV25wNaNAnYKMDBLNfHewejpQL3bt+evn/06PyOBX1wWj5Oz5/EeNlBzyULK+VJC3GpwiTveL9PXKrQX7PwWN0V0IMjzsRhGGvBsf/JVwufMWv6LRqMh2WNr3F0nfwWDXLVv5a34oupjTgt+uDBatwaPvQa/hw3f9XAuuM7DOqKq47imgvXlasNAI6XS+VTWzGsE1dfxSbho9lRXtLwH7/4H9NnvvqZ9I8+8I/Sxz7zscHB65WXXqlhV9X6gjMOxmWOg9zSUFhnDneY08SdpbcdOzY44+DpOfFHzjprsHYcd53OY1vzGQk78hI3Vdwmr625cbD7He9Iz8rYUy996aAB8civ/Ep6NK+3mMvH5T0N5mIgt2k0nW/LfHs4+7///ZQez60cXH+LnXru2G/L70B+5v3hKf5L2JNHcKMCS12BaBgsdcWni7eWbpY4XSXCOyoQFYgKLNy/5MFRud/Yg5OL2uo+rR7Uqb7yHa4YNMmv4bWY8HWmOtRWDfgwd8fFeA++mNrIRfWZN3BnysU4+YoTw3jPPMF3OqoBDmuuOLGaBnBneiYCxnvORtB5jsv7t77wW2nDUxvStd++Nhclpd/4id9IV5x/xSAl6uh8MMA5Qfua665J7/+L9w8fIYj/guOVD53+/f/z79P7fu59eWN12mYtnJtCbbwsFnxbMaepvk+87nXpkfykBBiaBEfzmQ9qT5yKOyDk+zDkMdg5p5+ens4Ng615fXt+oZEweErDF784eHLDGb/6q+ncjB3LzYOHTBNhXD7ZdWCaYy8G/lLFYW6zXO7P9U35XhSDpgGaRTi1mKcX78oXkOSbXu7Pl5wc3r17lmFDKyoQFYgKTFSBaB5MVLZwigpEBdZ4BXRflgc6mHIPrlz4Uqfczx13gKU61OjNpRYTOs5qMVVn1nlPqo38WZeevN28iblcFANv0vmrDjWg53JXDBzOs6YBjjNtJPQ0EaDFHBnzt//mt9N1t1+XPvftz+WDzEzAf6HxL/b8f/NfuO0X0v+6+n+lKy64YsHxHDUyK11z0zX5YDOl9/9lbhigUYAHJOF/7jiBEq9sp2w4JR05MrzAvnZJw5C5Mn9WLyRlETXt8k3GWCsGbqvmzksuSUe/+lW4JJxhAHuGB6mDreGP4zhgzfZ4bjSc/Xu/l57MB7cP/9EfpR/MNQzYREAj4dH82pLxZ+UmAs5dOI4mwpvfnO7P6zvmzkZwc8nDzXN0/q1znjYO/BfVbr89pW99a9AgwNkFucWHv/JhSLwPeJ2Cdk1YVCAqEBVYGRWY5GaJeKQjmw4rYxaRRVQgKhAVWJwKlPutPHhTnBgy6MV1H1gPsJyOYojFuIoTG5VLLSZ8Sqtp1zQcXzHoM8ca3qMNPdXp0Yav1hzbpdVyqeGT5qJ5OA3k5XDNAxzVwXZpPU0E+Kr+G/7rG9LnvpEbBmoP5Y3Bf6LzEs2DP/qF9Ouv+vX0nte8Z5711/f8dfrc7Z9Lv/UXvzVsEuB/2ngw0twhEpfbN2xP73nd0G81P+pxQeNACzhfkbyibybxVsxpjvN99s/8TLrnf/9v2yyYP1idS+Tsn/7phBc0N+cGwGm5gQB7KjcJHs1NBDQMUl4/6ZKGjO3JLzQRnrz88sF7fDQ3EnBWAmxcjgNShefmPK2my4c5LMkyX4c8aA7s358OP+95SxIygkQFogJRgWkrgJseHjhwoEtmuW6WuAtn0cUlX13vVZCjAlGB2VVA9zXHHaQ6LjKp4bpvrAeANb7DFUMslyMxjNdiYqy0mrZqwIe5K19j9uCLqY1cVZ95A3emXIyTrzgxjC/HPGu5IJ/StImAsVFnI1z3jXyWQdk0oCDOHEAjIP+/FM2D3z782ynhOvo9KX3u7z83aBrMn12Q4YHNNQ3QLIC958B70jv3v3Owrj84HzQSnsz364Ph6Qw4E4Hrg5UV9KN6jwP9hWC+rRgLQT8uW/3B25FPez/2ta/RFZ8OJ9bn1s76qZ9KeDldNBFOnWsiHM+XPeBODA995CODJsKWvI63ks2ErTfdNLik4cx3vzudlpsI+N14AE2EvA5z+g5z83a8VqwWe5DUUv/AL3T+ZU75xpNhUYGoQFRgNVUAzYOeJy3gxooHDx5cljMPXod7yOCzNiwqEBWICixjBXRfddyBsXKRMvk1vNxf5gGp8qkBvR5cuZpLLSY4zlRHc1Edlze0yK9p1PCl1nbzJuZyUQw8N3/OHeOznic0qa+5MA+M10wbCWUTodo0oBgaB3jN2aB5gJPe8d9p4HgMoNiOjTvSb+z/jQHy9te+XUaGq64unE/PUxpOEl4CYME9DnQiGtvhDuOkJ/Et9Z7/nvekR/LlCvf+2Z+lY1//+rwkGgCws3PDAGcajDLmgyYC7NS8RBycjfBwbiJ8f65hwEsaBmejZAyGJsIFeflIbh488KY3pfvy+vZ9+zC0wBhjAZg3yvlgfBqs1F/K7f1b890jcGnC8eO5o/JASnM3qlzKHCJWVCAqEBWYtAJ79+5NV199dcJlCK1Gbly20Fqx4EUFogJrtQK6/8oDN8x1VrjuS/MgULURi3EVJzZJLi4mdJzVYtY0HF8xxGDuihPD+Ky1NaZqA2fNse5M+cp1uM4HWpxTDXca8FM+NWq4aoCjOWK7tLKJcN0d15WU8dv5ROyBzTUNdm7cmbZs2JLe9dp3pbe99m2DIc1bc9T8avMEfyVe0jA440CTnivDgjdsFKaFGMVzMcB3ODV3vuAF6aJf//WB7NG5sw/YRKg1DejLXFwMNBJ2zp1NcPd/+k/pyGc/O2gW7XrkkfmzEHBfBFymgrMRzvy1X0vn5PXjaCL88i8nXPKyOe+IlubmMg1W6i/bNk6fxdMU8FSFfK+D/fiPGLbRUNizJx2Oexss21sTgaMCa7ECe8wZZtPOkw0ANgRa9Milb4tPcKICUYGowFquQLlfy4OjFpxc1Ef5iut+/LgDLNWAJnVacHLhV4uJsdJq2jUN5WtMhyuGuOSrNnDWRfnkYrwXV31qQ8eZcjFOvuLEMN6Ti2qodo8G/FRHc8FYaX9z298kXKpwkuGMAjYHThocAv/6tf96sLJz0/B+fG999VsXMDVvDPA90vyAM0flk4tx8lfCJQ2bNUldR6I0h3MS5GDpeK3YOD00EWBcDjbmfjhfDLXEfs6/+Tfp6M//fHo0HxDjDIRjebn7u99NO4smAs9GOCtz0ER4Ip99gIsn7n/jG9NRcyZCS+xajsBXlD34YL7OI7dL7rxzcKlH+t73hmcg5Gty0EzYn59qkXINDl988YpKO5KJCkQFVnYFLsufLefka/rPymc0fTJfDvX6RT5Fnw0ANgRaqgPuvvz5hrMWwqICUYGoQFRgYQV0f1cPdhyuGFTIV5wYxsv9+3EHWDUdhyumuWhMxsN4zVSHuasG/KijXODkK04M4zVc9ZdCG7nUzOWiGPxcjks1T82Feehctm3YlrZu2JqOP5PPqoadll84iwD3McCBHk91z6ul/dprfm0eUu1azFm8n9QuL2nAvRH4mk9qEVY2fPGLX8zzOLkqDmOymofjtWJOD9rOX2Ny3fk73x4MDQTYfZ/6VHo6NwnQRNiRmwi8LwLORBg0EfISGO6HgD7Tw3nH8t5f+qWTmgitsbPEirX9Dz+c0p/+aUr33JPS85+fr+M4c3i2AfDbbhtewoB7H+QzMg6/8IUrdh6RWFSgpwJ35qeHnDv3yBz1+9rTT6dLcOlO2KqtAO53gPsY9NihQ4d66MFdwRU4eHBPvmzlrpzh8O94794N6ZprBidgruCsI7WowOqqgB4YauazwvVAbZz+YsfU+FyvxezJG1pOx2HgLqb2KH2MlVbLpYa7OTkMcXo0wB+ns+8D+/J/hPPl6RvzMxjzCdWD18mHxpBaYP/jLf8j/fgP/fgCjBsuR4eBPy4/anI5io/mAWyxHvU49owDd3COhFoPiB3PaToe4pTW6lvTc7hiO3BgnO28iy4aLB/9xjfSfX/xF+npm29Op9511/wlDbwvAhoJg0c95vFz8gu7Hk/k/0yhiZD/T5+2ytkIGmcgvkp+HN69O+3H/STyJQqHn/3sBVnvf9GLUvrSl4ZnIuAeCNE4WFCf2FjdFfhE/p0uP6BfYB4Nu7pnuf6yx5kHPTdLRIWW60kL6+/diRlHBaICa6EC5T4vv0sVJ4b59uJ6PKAHZE5HMcRiXMWJjcqlFhM+pak2xqhf01A+ufBzuGJLpY04tdwxVlqNW8N1Tpy/YtAnrhrA+f4rn1yMO/z6B65P13/z+vSBz34gP14vk3B9urnR4YKzESDWaJoj81MMMsRdfhifhL/Y90Wo3uOgTBYT0IlhG9aKteoNVU/+2erfmo/juRg7fuiH0va3vGWQEJoID3z60+l4PgNhz6tfnY7mu25vzM0CnIGABgJfW/N/sp6T/6N1bsbQRMhP70gP5B3VR1bxqa6H870MnB0+77y0H49qzPeJSLmxsj+fvXJ4Ea5PdrEDiwpEBaICk1RgkpslLueTFiaZY/hEBaICUYGVVAHd7+ZBnWLItQcnF366/86DMeCqr3yHKwZf8mt4LSZ8nakOtVUDPszdcTHegy+mNnJRfeYN3JlyMU5+De+ZJ/RUh9qqAQ5r/rs3/m764N98cNgs4NkFIMDmbnR46qZTB5vvfvW700f+z0cG60/84In08NMPDy5p4PjmZzYn3GixfErDwEF+uPww7PBa3r18as/6vggLzjhgEJnrgl9S4uWkRuGtmtTQZauvy8dh0HZ4a5zt+SyEc//Fvxik+Ogdd6Qd73//4HKGfHZLOpabCBtywyDfLnDwqMf5SxoyhpNGnpcbCXgq58O5eXDPKm8i5GksNJyFwBsk5j+geA75wvLEVlQgKrDyKjDp/Q4wE/quvFlFRlGBqEBUYOVXQPfFeUCHrHtw5cKXOuU+/bgDSdWhRm8utZjQcVaLqTqzzntSbeTPuvTk7eZNzOWCMYdrTHAmyeWDX8uNgvw/zg9em5doFvAxinLVKe51sG3jtrR7026EGdhH/89H59byP4fz2Fkbz5rfvvyCy9NLLxw+ta98SsM8yazoHDHM91lxYhjX+XPuwHv45Jb3RYBO7yUNgzMOKAgBmibai7XqUVeXzhfj0+TjfF0cx6vF3n7hhYOcNs5dioBLEuD/dG4UPIQzEbLj5rx+0iUNGXtubiKg8I/NXdKQH3CYtq3isxHyb2++0UO+08O23C7BkxawHRYViApEBVZ4BdgA6L1ZIqZF3xU+xUgvKhAViAqs6AqU+948OFKcGCbSi+v+/rgDMtVGLMZVnNioXGox4VNaTbum4fiKQZ851vAebeipTo82fLXm2C6tlksN78nlAzd+IP3eZ39veKPDx3Jk3P8QNzyEzTUNTtt0Wvr9N/5++uq3v5r++LN/PBzLP38wOF8cdDoMh56Zu1viP3/VP5/n6kpPEwF+bp6KgcMaurmXGq38SS9p2KzJaUIIDGvFwFUtbMOc/3Bk4c9WX6fXirkYtRxbNUvehtwA2HrNNYPJPZObBI/kJsIjeWtTXneXNOBMhPPyOC5pwKU1D63AsxH246kK11+f0qmnpsOvfOVgbvyxH9d742kL+T4I6ayz0uFoGrA0sYwKRAVWQQXQAMBlCD03S0SjYTEaB0dzA/Yr+b4aL89PmFC7Nt9f5or89ImwqEBUICqw1iug+9XjDlIdF/Wp4XocwIOxUXynoxh8XY7EMF6LibHSatqqAR/mrnyN2YMvpjZyVX3mDdyZcjFOvuLEMO7m+Xt/lxsF+VTw3//c7w/PLkDDAE9IwH9053oAp286PR/+P5PedeW70r7z96V9F+xLP3rBj2ZCmm8ebOR1CxljEwHjbCTAb5xpEwHctXBJw2BPRAvPIrRi+maO8uWYLnt8W/NxvNY4zhf5OtxhOjeso4mwWZoIODtmcF+EvJOKyxd4NsLgKQ0Zw+UNaCSgiYBG2Iq5pAE3PcxPl8h//Wl/PtPi8Lm4e0NK+/E4RtwYEY9nxOUKP/zDAzx+RAWiAlGB1VSBa/Ln9IEDB7pSXqybJT6Qz1zDoynV9sR9Y7QcsR4ViAqskwrovva4A2Plojzk1/Dy2IAHpMqnBvR6cOVqLrWY4DhTHc1FdVze0CK/plHDl1rbzZuYy0Ux8DD/v33ob9MN37oh/f61uVngbnSIpkE2NAze+ap3DtZ/6ZW/NFjqjzf+5BsHm3rmAQBtIqDJgBdMc+H7MBio/NBGwiybCAjX837W+JzPqEsaFtzjQINyzg6jMDllAoq79VZ/F7sVczFqebZqOp6bn8PQRIBtyTuo3/noRxPORjjtO99JO/IjDV0TAb/juKQBj3o8kn3xlIb78vopczp5dcns8DnnpP3nnz989OLhw2k/H0X3eG6F4GwEPJ7xJ35i8DjGhCZDWFQgKhAVWGUVQPOg50kLcbPEVfYGR7pRgajAqq6A7oPzoBgTmhWuxw08CFRtxGJcxYlNkouLCR1ntZg1DcdXDDGYu+LEMD5rbY2p2sBZc6w7Uz65f3DrH6Qbvn3DoGkw/xhFOs81C5616VmDswt+9VW/mv7ZK/4ZRwe/N5yrarvmAc9KwH0N2DRgzZwGgjDH+YDFyqRNBNXWvBVnbsCYH9aVr/mN4peXNHSdcaABkQBMgw0R/7PV1+k5rBZ71nFqsf0s29Dz3vSmAfHRb34zfe8zn0nbc/Ngc34qAX7HT83rekkDng7CpzTgkobH55oID+X1LUvZREAsPBv0vty+wFMU8qmzA8PTFvJZCOnSS9PhaBoMaxI/owJRgaYK3HrGGekyNB8Lw3/dl/q/7JM8aYH3RliMyxaKksRmVCAqEBWICsxVoNw358FRC04upJSvuB5LjDvAUg1oUqcFJxd+tZgYK62mXdNQvsZ0uGKIS75qA2ddlE8uxntx1ac2dEr70Dc+lHAN+IduyEu90eFcowD8UzaekrZv2J7e+Q+HZxf801f804EMchqXI/K4+hVXp73nDf/he/mFly9IoUUDDq3zAbeniTBK28XU9wG+nL9ygbPmyicX4+SPPeOARDjRVJRYbdnq7zRbMRcD+bT6t/Jqc5wU35EPuHewiXDVVemx3Eh4IJ+JgIbC6flshJ1FE4GXNOBMBDYRcLnOfTO4L8KuHHdnfrTksXyTx6OmIXEYlyLglW1wX4PHHsNvX0q7dkXDYFCV+LHWKnDukSPpRjOpS3jGjRkLqL8Cn8iXPG3AvVLElut6fjYA2BCQlKqr5NK3SoyBqEBUICoQFViUCuh+vB7sOFwxJEO+4sQwXh5jjDvAquk4XDHNRWMyHsZrpjrMXTXgRx3lAidfcWIYr+GqvxTaH7r1Q+lD1+VmAf5viaci6I0O53Yh3v7Kt6fbv3t7evn5L0+/eMUvIv2BIVeX46h57j1/2DiYxTydBnMrl9pEwNhyXtKg7z1yQb0WNA4A0nSSxEoB4uWy1dfptWKIuVRxyvktxvagiYD/3F955aBxgBjfzWcj8JIGNBHcJQ3AyksaHjEH/tCr2c7cNLgwNyPQhNiQb/qVWwLpK1nj7Lx8IDcleJkF/Qc3QMQTFGBxlsGwDvFzTVbgKjwlJGxdVYANADYEWiYP7r7cdMVZC2FRgahAVCAqsHwV0OOIUQeGzNDxFQNPdfTYwx2MKl91VKMH13jQZkysO6tpq45qOL5iiMHcFSeG8cXS/sPb/zA/7z6lP/zbvMTZBThAKW50iGYBjMvBRv6BnHSeLkedD/w4J8WJYdxpAHd8xcChjmoA1xyxXZo2EnqaCNChtsYkhnHNkfkBH8XfcN111z0DEkyJQ2ShKDG37PHVRKnVirXGcXqI5XCHMa+VsMQZCA/kJsK5L3tZenLuUY94SgNvrohDeLzQRMC1J3jxkoYH8/rWETuzO7LO+WwaZC4ujbgtv0rbcuhQCcV2VCAqEBWYqgK4VOGOu++2Zxws9aUK5URwv4OeJy3A/9CUn5N4qsJf5zMwnsQZXWKoxXKdhSFprJnVgwf3pI9//K48n+HzuPbu3ZCuuWZw5eaamWNMJCoQFTi5AnpwxFGHYawX1wMyavfqzCqmxtd1pz+LvGvznET780c+nz7/nc+nD1//4eGNDnF2Ac4ywH83xd72ireld/zkOwQZvVrLpYa7WiFCDXc6NW4Ndxq1WY1rIpR+NW2H1/IbzP/aa6+dbxxokNaD6WkO5F0Mh7kYyNVxp8F0/it9HWciwJ7KjQT7qMc8hkYCdovwlIYf5CXOHtCzEbZnDTzFYXCmQR7H3+Yd+eVs49VXp03ZPywqEBWICsyqAiu5cYCmQc/NElETnHGAmyxOatE4mLRyfX7ROOirV7CjAmuxArWDox68xnUHY6hhjT8LvBaz9t7VYjqdGrcXd9qsy4e//OH0+Ts/n77wnS+cOLugSP6tV7x1UEM0DUqraZc8btf4Dp/lPBmfy5q2y4M+btnTSKhp13DNcUGL3R10u+TcgbzzdRj0HO6waeI4vVpsN8eVjvHyATylAU2EJ3LCx3IT4f9n712j47iuM9ENNF4EQJAE3xQpwaREiaIIUrTlh2zFdOI0o8Rm4jzHHtLmJJNJZmWWZsa5ylrOWrPG6/rHzFzlOnfmxj8SR7ZsYJzJ8tzYoZKMCXtiJIztxA+KBCmSksWHxBceBEAABEA8efd3unfzoFjV6AIaRDf623ahqnbt2ufU1xDY56v9KNNjBFdbNILVRQCJAJKgVveD+uUWrR5r1R4RPyAO8H7rLd0oRIAIEAEikCIBDiphGidlgZ0W+JtDBIgAESgOBPx1gr8wiqP3bfHU5ie4frEFWZS9rzcf8BdHHzUm/IRJlG/fT77nHfT9x29qVIEWOvyTH/1JZKHDt295u7xjyzvEJwts7j5WQd9hz+zrouzD9Dae3W/j+nrTwSbMB/Rh9r4ONuYnygdswuR+pTRE1jgIm5T/EHY9+MDQz0cXNsZ8fYbNx+a/FPY+iYDnmVLyoD+d0lChx0YiIJ0BUQWVqlunJAKOUVsENUXQ6jGbTGsOLyMOsiHEa0SACCw1BOZa7wA42L1xMKkf1mSxihl8fpzbaUsEiAARIAJzQCC4TrDFm683HdzH1ftrG1uMR/nxfcPGxvX1povyAX3UmLgWlCjfUT7C7H0d/Nscw/R/cvZP5PM/+HwqBQG1C7BZKkK60OGaijWyrHyZfP7X1E7F95PNN2z9eePcxxznQfHtfdso/Vzn4vsO84F5hen9ecDG94PzoMQhEXCv7998+zrYmH7WbyjBG3Gz/1A4h8xHh/vv1zgYa6kLiISqdLgsohGGlETo0joJfqtHRCUgRwX/fU7oNjOjVhUhMqV+SB6EAEMVESACSxYBIwDiRB6Yrd0bB5yfQatbtL6lEAEiQASIwKIg4K9pZlukhtli0lF6f71ji7Fs9mF+fB3uDZuj6XA9akxcC0qUb98H7rG5+/b+mEH95y8pAaDc+Od/pHuESePtJfKoLV9aD9dWrNWfIv/qPf9Kdjyww23mJ5tvd5P+MFuc+/b+3G3edk9w79vimtn7etPhetSYYXrfh+/bt4Xe5u7rTYfrvh9/LrgWFJ9EwLXZUhqifJs+lDiwi8HB/Qewa/PRhY0T5g9jhelz1dlcS3EPEqFCSYQKJQ56jxxxHRtWaKvH9dqlAeQB/ntFzRFf1unJCt1QH+GaboO63W5okHqtGk4hAkSACJQaAiAAkIYQp1giOy2U2m8Jn5cIEIGliIC/1vAXb2F6XwcszD5KH1wH2SLQtzcf8BdH79v6c4kaEzZh4vvx5+L7CZs3fMH+82c/L3/6/T9NkQSowO4XOkxHGYAw+M33/KYb/iPv+Ijb4wfGyOYbNlHzi9KHzRt+oiTM3tfhvrA5+ljFnUuYva/DmObfn4vNA9ejxCcS5kIizCAO/MFtwOBEoZ+PLmyM+foMm4/Nn/sUAq7d43PPuRN0aZj+rd/KpCngv2NfkNIAEhDEwYO61enWqZumIUmPfnmuUTKCQgSIABFYKAQWu6NC2HOh6OH+/fvDLkXqWjRKaz7FEiMd8wIRIAJEgAjcdwT89YYt3DCJfOn9NZItAn3fGMvG9fWmm8tcwsaEnzCJGjPo40/f+FM5fvm4HL9y/N5Ch0YWVKaiC37zXb8pv/COX8gMhzH85wn6NsOwufg62JmfKL3vG/aGOY7DxLf3bcP0UWNG6cN8YA6+vT1PlN73ARt/jjgPylxIhIrgIHDqT9IGCdPFsZ3POHHGtvlyH41ArUYbWG0DpCoEIw6uqw5FFUEgYAOxUKP31GtdhC16bK0e+/W4kiSCorBwkjx9Wj+Eaml75JGFG4SeicAiIbCrr0926e93sQhIgDidFlgssVg+Wc6TCBABIhAPgeDaxBZ1uejNFiP69r7eXzf5C8Awe18Hn+YnF73Z4r6oMXEtKEHfX7j2BfeG8cUTL6bIAqQiYIGRrllg96+rXCf/8l3/0p3+/Nt/3u3Nl83Fzu0e0/vzwzXDxbc3W1yPq/f9m2/4CRPfFtfN3tebDtfjzMX34fuO4wP3+X78ueBaUHIlEWZEHPgTMofz0fkTNn/Yz8dn2L2+bx7PjgBqG9zS1INlSgakiT93U//mzXJzyxZHEkCxUlMa6tTGJxGsS8Mmj0SAj55Aq0fnkD/mjEASec7f/77Id74jsk6TR0gczBnLud54ZcUKOd7TIweqkNSTku/q54K34Y8mUv3fTc99aSCAdotxOy3Mp95BaaDKpyQCRIAIFD8C/vpktsWrb4snN3tfbzpcD66nbBEYZR9H79v6c/HHtPFwPSgvvv6ifOEVJQ1AElihQzNKkwa/8c7fcJr1VevlwN4DdjWzHrRn9ediOhhH6cPm6NviXvPj600X1zfsoyRsLr4O9xmO+ZiL7wO+7Zl8velw3Z+LzQP6MPFJBFz3UxpCuyr4g5rDXHWw9ycX9/4445hv7uMhgLoH13fuzBAENQMDjjAwL6hnAOlUGwiiDVAXASQCSnZZlwYjEaADkYCUhgH13U0SQZGYp5w7J5pQLXL2rBaRGZOkhm616YKVcv8ReHkC5UPvSiGG0d+dHY8WGgEreGiEQC7jma3dm8s9tCECRIAIEIHiRMBfy/iLtzh63xYo+H78dZYtAqPsfb3vI47eHw9z+eLVL7pw5C+cUMIAZIFFF3jvVJ584El5ctOTmegC3AcxXzZv6MLm4utgY3P39abDdfOL44X0HfSP86BEzSVM7z8P/Ngz+XrT4XqYD+jD7H0dbMyP7wN6Hy+cB8UnEvIWcRCcBAYNTtgmEqbPVWc+uJ8fAqv37ZPe9nbnxIiCKI+4vgYFFpuaZFoXszfTrR4TemwkAoKN8XcDrR43etEIIBH6VF/FlIYoeO/RJ0cUyYsXRa5f1z/I+ItMIQJEoJAQMALACIFc5gbb3VpgFlELFCJABIgAESgNBILrG1u8+XrTAZG4en/95S8Aw/z4Ooxl4/p604XN5Quvf0G+eFxJgym9ii/9eK9i77TSpMGvP/XrYhEGQR84N/9x5h30Yz78ed8v3xgnau64FpQo2yi9/0xxnhPj2ucf5gPXo/RRc8E9QZlXxIE/kO/Yn5jp56MzH9znD4FlSgLkKpsPHxYUV4QEWz2ijSPaPYJE8FMaQCRUqc5SGsb0y/K06noZjaAoREsSb7cvXEBckOaKrGRbtmioeIUILCoCc+m0gPoIR48eXdR5F+LgQ0NJGdfCO6tXtxXi9DgnIkAEiEDeEPDXQ7MtDH1bTCDM3nS47q/LbBEJ/UvnX3KL/U888omMD+h9/+bH18HmpUsvufaJXzyphAHaKOKL/x3dIGnS4F+841+4U6td4E4ifvj+bUx/3rjN5h5mi+tx9Ln4hk+bSxzfuM/3b/OGPkx8W1w3+yh9Pubi+8CYsz2nPxebH+4zmVPEge/UHAUnBn2YLkofZWv+uc8vAkYEzObVJw2CtiARIJUajXBHSQL8PRlWEqFMj5EVbtEIfkrDA/rFuVavDeq9XSQRFImAfPe7Iv1adhKtL9dqxdkrVwIGPCUCRKBQEAB5EKdYIuYNe3ZauPsJJhJJ+cY3XEaWPPtskuTBXWh4RASIwBJHwF/72IIOjxxH79viXvj50uUvaTEzkS+f/rIIVnr4Uq5v7050npBPPPYJ2bN2D0yd2Li+n5PDJ+WV66/ISydeSkUVoEp6SKFDEAbYzEdwfRi28EyNmvrpj2k+cMX3Yz58W9iYva83Ha5H6cN8R9n7PmBj/n296XA9yjeuhUmUfZjeHxO+bFxfbzpcD/MBfZi9r4ON+QnzkXPEgX8znJoEB4N+Pjrzy/3CI7D9058WtGa0lIVRPfYlG2ng2+HYJxFwPqXkQX86paFCj41EQCSCpTQgGgF/z0aVREBKQ68eV6fJCD0sOUmCMMCGYnxbt4pocT6pry85HPjARKBYEJhLscRsnRZu1dXJP2hNk59BcdS0oMYG6mo8XYG/lktLxsaScvy4yA9/mPqz99RTiDpYWs/IpyECRIAI5IJAcO1kizdfbzr4C9OfvH5Sfvfbv5uqO+APii/h+hbvxPUTcuLqCfnDD/yh7FmfIg98P1+68CV5qeOlFFmATFk/WzZd6HBj1Ubn+c9//c/dHj98H/4c/bWjEQCZmwIHcX2E2fs6uLe5ROmj5ufbmw/4C9P7Ohvze1e+J+NT46nFu0ZllOF/+u94RaJC6irqZO+mvTCdIVFzidL749ocfR2cmz6OD9zn+wn6mPFNxDfEjRB/sJRmpsN86MwH9/cfAUQe1GoqggmIBBAISGXINSrB7vX3wZSG7s98xkU2LQ90aQCRgJQGRCKg1SNSGvCVub/EohGSb70lgm3TJhHtbNGmxRCTIA2W4GLB/z3hMREodgTmWu8Az233BjEIFuQMXi/28wsXUoSB1twVbFevprhS/bNHIQJEgAgQAUXAX5PZ4s3XAaSg/suvfHnmYt+QRHoB6hJg1aeRB//+G/9ePr7r4/KJPZ+Qlq4WF53w0qmXUvfCNhBdsDyxXLAd3nsYq2DZ37x/xvz0jnvm4utw7K8n45AIvh/fB/Tmx8fFMMH1OPp8+f76ua/LlYErMnVHATfSAAcqNVU1sqp6lYxNjcl7trzH6cJ+RM3F19uz4/58Pyd8Go5B36ERB/7EcLOJf3M+dOaD+8JBwBEJShrkU0AiJDSdAdJ75IiLckCXhvqILg0gE0AkgERI6HFJpDSgrsGwxoK9/e3StmEDoEq9gitXipffplN48GfJINCrv/PF1L3CCIC4xRLxgdq9JfPh6oNO6xdX/FnDnzdwo5XamgdlXdg4ppR+C/isRIAI5IqAv/6yBR3uDepPdp6Mdql1ZFwxQ/27CxLhyz/6snz5x0o0YE2LUGBcT61v9Y+zHqtsqtokezbucVuyOZlS6k8bN9tcYGx2dqPZB9eZ/iLYbP2978d84Lrvx3z4trAxe19vOlyP0s/Vd9dwl9wYvSGralbJsspluo5JKKwKrP6/uqJaaiprpLK8MjN3mzfmEiVhc/F1uM/8RD1PlN73Yz7gL8weukzEgX8jbjDxb8yHznxwX1oIWPRC7XPPuQdHZMO19nZXGwEkQm0EiYC/XcFWjz2qq1lCKQ1J1DFAJwXtXpEhDUrr14NPW6IInGpslF19fTOeHm/bizEsHwQA0hCw5Sql2mnh4Yfb5OGHgVJSTp8W+frXU8QByAMKESACRIAIRCPgr8v8BTDSEGYVRHUh8sBkUA+UyHVEQZo0AFkA+cSTn5Bt67e5Dedh44bpomyz6f01qL94xT1BiRozykeYva+Df8PR15sO1+P4vj1+W2oSNdK8vlne/9D7M77hJ0x837g+2/P79r5tmN5/Hvi2Z4rSh/nAfb59hW+EixDfIKUJ18W1NV/cEwE/RQIkQqeSCMuUPKg4dsz9/QqmNKAuC1Iagq0e8XduqIhJhCQWTW+8oWEVXRoidluSN25I25o1qV+QM2e0iqT+VcerORzv3MlfnPuIwOaBAdH063vkUa5u7sFkPoojmtNfhlfPS0BQ9HD//v2xnqRFa8GUbrHENnnyyaSgJuzQEL4wxYKOxkSACBCBkkbAX6+1nNCUg7lI+u/uA1UPyMef/Ljz8MEnPpjxZOtEf5HqjzvbYtS3hdMwe9Phuo2HY39MnAclyneUD9/eHzNM7+swrtn7vqG3OZr91PSUJMoTsrJ6JS7PWFObj2x637/5do5Cfvi2uGz2vt50uG5zxPFsc/F9wN78ZCIOoPQd4hwyH13KA38SgewIzCARDhxwNRZ6lSQAobBRF2nleuy3erQuDYhEwAuq20ocgEDoKca6CNeuiZw/L3LxooaJaZyYHidrte8EyAIlEVwCMOD76lclCZsPfEDatIAa5f4gcACFKilEIAYCIAHidFrIViwxxrBFazox0aZpCkn9EoMvMkX7GJw4ESACRGBREejoDES7Yd2Kv6moWeAXOQzM8tCeQ7K5arP81BM/lbliaz9/cRm1kDRb3Ozbx9H7tr4ff0xbuGYmGXLg+7G5+D5wi/nxbaE3e19vOlyP0vv+z/Wek/KycpmcnpSh8aHMPeYnykeU3vdt88ZcoiTM3tfhPvMTNWaU3vyE1jjwb7LJ5aoze+6JwFwQyNRY2LfPEQeVTU3OzTiIhHSrR02HzXRpMBIBumBKQ1FEIqCE+IMPphJ9b2nvHF+C36JBJlCIABEoaATm0mnBaiN85Ld/u6CfbaEmF/xTt1Dj0C8RIAJEYKki8KmnPiWf6/ycDA5opCreeYAsQO2CQKHDsOf/yZ0/mVnk4nrYQtfX49gWkjgOW4xCH+bHdLjury1n0/vj4V4bE8dhEuXb9+P7CLP3dRjD5ujrTYfrYxNjMjE9ISPjI9J1q0uOvXlMxqfHpaK8QpZXLZe9G/eG+sjFtz9v2Ptzx3lQfHvfNkzvPw/82DP5etPlLeIgOGGeE4H5ImB1EeAHBRYr0wUWp5VEuJlu9ZjQ47BWj8GUBmRRVxVgSkPbRm1rgy1EkufOiXzxi6nog1/+ZWl74okQK6qIABEoNASs4KERArnMD7bDGk3U9Ku/mos5bYgAESACRIAIOARaLrVI64XWFEmACIMB3SyCK4cUMCwQbWEIh2ELxmz6sMVolL3vGzY2rq83XZQP6KPGxLWg+L5xzfxH+fDtzRb3hel93a3xWzI8Pix9o33y1s23ZOD2gCTKEo5MgP6V66/IgysflJ/e+tNwl5kHjn0/NqavC9pHzR12QYmyjdL74wbnMq+Ig+DEeE4E7gcCwVaP+Bs5pEQCSAQ/paFa9aiLgEiELXqMLg2oAdNbLCkNqGmwYgX+OrK+gX5uFCJQTAjMhTz42h//sfzy9u2y4dFHi+lROVciQASIABG4zwi0dGpNAy0A1npGCQNEF+DLML7kgjBIkwbve+B90rSmSaYGpwSpDJfHL8vgFCoipqQh0SAXezQNVsVfLOI8uGCEDhKmNx2ux1mMwt4f1/z4OtjMpo8aE/eGie/ffPs+cI+9pQ+zxfUwfVWiSlYvWy2VZZWyvHq51FXWSUWiQkYnRmVgbEB6R3tdm8ajbxyV/Q/Hb2kZNibm4s/d5g19mPi2uG72UfrgmHOKOAibCHVEYDEQAIkAQTTCHSUJxvR4OJ3SgEgti0bwUxrQ6lGrCMig3lsSrR71WSlEgAjcfwTm0mnhf/7u78q/+au/mjHZYuwyMeMBeEIEiAARIALzRqBjpENOdp2U1tPp6AJUDkcqgkUXpPeoWbClaotMDUzJ+QGto6UCkmDnslSRbSMPoDu0+1BmEWyLaNgHF4zQQcL0vg425ieXxahv7/sxH7geRx81JvyESZRv348trv/24t/K9aHrrn4BfDVUN8iWhi3yzEPPONdv3/h2mbwz6RbyT295esZwx946JpeHLsub/W+6Ggjm33zDOGwuvg42houvNx2um18c+75xHiZR9mF6jJlzxEHYYNQRgUJCwCcRMK9Ln/ykoNUjorTQpcFIBEQiIO2rMh2NAPZsVEmEbo1E6NXj6jQZoYeLKm1TU5Lcswd/SaSNScCL+llwcCIwVwTm0mnh73//90u408JckeZ9RIAIEIGliUDL5RbpuK7tflEA0aIL7FHTZMHB3Qfl0o1LjiywS1F7EAaQprVNbjM7fzEKnS1Ifb3pcD2uPmwxGuXH9w0bG9fXmy7KB/RRY+JaUKJ8w8e5G+dkYmpCxqbG5PYEPgRdLySqXRrC2Z6zsmPtDqd716Z3ub35sjk+8+Az8s0L35Qf9/xY+kf65XT3aXli3ROR87P74cx84DhM7+t8e//ZoZ+NSPDtfVtfP2vEAQaiEIFiRODBz37WFVjsbW+XMSUOqrXVI8oLBls9WkoDIhEspQEEbn8BpDS0/ezPFiP0nDMRIAIeAgcPHpQ49Q5KvdOCBx0PiQARIAIliUBLj6YiaJva1rMaXYB1akShQ0QXIGpg3+P7HE4v/f1Lcqnnkjue7ce+HfuymvgLUlu8+jrcHKY3Ha779r7eX4z6i9Qo+zC9r8tlLlFj4t6gBH0/tuYxwfZK5ysyNon4ZpGayhpX8BDHvr3/nL4e6QuVCZRzF0HbRrtm9v78YGO4mB10ZovjuHrfv/mGnzDxbXHd7EMjDsIcUEcEihGBYKtHPEPvkSOOUEA0Qp0SCmEpDSATQCSgLgJaPuYjpWE9UijU163du0UbL8p4gUQ26FQoRGBRENjV1yep7MqZw69eYhE2c6l3YESD3TsTIZ7FRaClZUPmltZW7WYjVzPnHR312kJzsztvbr4lu3ffEuwpRIAIEIH7jYArdGipCCAMUmvU1DTShQ5BFqxIrJADTxzIEAY2z8M/cVjaz7RL+9l2U+VlH2eR6tticFvsRumjFqm+vfmAvzh639afS9SYsAkT87Nn/Z7M88DO92OLa7M1PzZ36GsqaqRC/5cox+oiJb692eJKXN9pdzPmF8e33R+2t7mUHT16FC9hKUSg5BAYuXRJEI2A2gggEWqVRAAPaCkNri6CnkOH0Jw63QbSKQ09elwTY+G/VkmD9Vo1HcQBNtx/U7dl6gPFGi3NQlWUAkfgNU0heTRx9w9+gU+X0ysgBFr074ARArlOC9EKS508+MIXknJT/yB+5CMib3tbW67Q5GSXIgUeDthO6PmPdLP/jpfr8WMzbA4e7FTcO2foeEIEiAARWAgEMoUO/egCK3SYHhBkAQTRBcEUg7TJjF0UeWBRBhahMOOmOZz4C13/9nzpbTHu+8ZxmP8wXZRtNn3UmLgnKP6YP7z2Q7lw84Isq1gmP//Yz88wRTrDmzfflIuDF11rxkPNh2Zc9/3YhTAdrkXNL8o+rj7KP8YmcQAUKCWPgJEIy5Q8aEjXRQjt0qBIgUTAdlsX/SiwOKYRBENZSITVulhY65EGXXpv2NfRyqNH9QqlUBC4oh0tjvf03POP0/bychIHhfIhFeE8ntdIJqQixBHUSWjO8jcmjq9CtP3Hf0zKLX3B/4zWl6quzh9xEE4aAIHZiQNYHTz4YyUPUH2MQgSIABHIPwItF5VMPtWqcevqG6kI+NOUrllgo1kqAs7jLvaRsmBpC7mQDTbmfPZxFqlxbDGnqAVtXD9h9mG6bGOGYfRPV/5JTvWc0rToO7Jz3U557+b3OjPorg5edZ0VEAGwrm6dfHDrB+/5fmk+o+YSpQ/DJco2rj7om8SBfUrcE4EAAohEgEzqwr9cjxGBgBQGbDhGigOiEfDOCiSCiyQI1EVYpfeu9kgDRBp06xYmFbo4YORBGDKLoyNxsDi4l8Ko+/fvj/WYIA1AHlDiIdDSslIjPFbpTdrW9h7R5OGMIOIgKFeVrLmuuIMmphABIkAE8oNAyxUtdHgtotBhegg/uiAuWZCfWebHS9xFaph9mM5mF1zUQh9lnw992Hg2F9t/7ezXpHukWyrKK6S8LJVbMjE9IcMTw1JfVS9bV22Vh1Y8JI+uedRuyezD5himww1R+qg5htmH6WbzzX8RMx8XD4jATARsEY9Wj5BxJQ9GlAgo0/09KQ1p3SZ9m1intkhpwFanpAEKLeJPxw3d0LUhSqZPnpSE3kMhAkRgaSMAEgCRB7kKiyXmilSYHXqW9+uGugYQIwlsn9Le/QlCAfdc0y3w6u+uEY+IABEgAjkjkCl0aLULrI1iumaBOUJnhEN7DrkaBcVMGNjzROXXx9H7tvDrL3Yt7x56WzBH2ft630ccvT+ePyaOTR5oeMDN5db4LVdEceoOwkk03bmyTtYsWyMb6jfII42PZOoX2LxhEzYXXwcbm7uvNx2u+3NcCN+MOADKFCIQEwFEI0wgEkHvQ0qD1UWwiAQQC5BR3WCDr59hReBUPUOYrjADjkU9YcTBosK/5AdnvYOF/4hTEQeICQMZgGKHRgT4pAGOcR3bTGluLmPEwUxIeEYEiEAMBFyhw1c1FQFvkFDoEFtAQBZAQBiUkviLXXvuMB2uxdX7C2bzHddPPsY816MtHDXaoEz/V11RLY+sfiQznTD/+Zh31HPmyzcjDjIfIQ+IQO4IIBqhKh2JABIBBEH3Zz4j9VojwVIaQB7gayqIA6StUYgAESAChoAVPIxTLNFs7V7zxX1cBHySwD+e6QfEAYUIEAEiEAeBlq4WEY0oaD2jhAGIAmujaLVYVdW8oVma1zeXHFng4xj2xtzXwdYW17nozRb3xXnrDnvfv/nxdbCZTR825mNrH8OtoeL7N9++D9xki/0wW1yPo8/FN3zaXKJ8kzgAShQiMA8ELKWh4atflRv/7b9lWj2uT3dpAHEwEPC/Ts/fltYjINYF027eLNBTiAARKA0EjAAwQiCXp4btbi3IupSLJeaCQy42hw7dVKwqNS0kPgFw8GC5Fkb0vunnMiBtiAARKFkEshY6TP8psVSEkgUp4sGjFqlx9L4thrEFcL4WzL5/841xwvRRY8I+TMJ8wM73E0YiwMbmEuUjSh/mG/7C7Gfo2I4RMFGIQH4RGD1yRFZ87nOZNAUjnDEKIhHW64a0BhyDvcPW19AgQ//hP7g6CNWsdaCILIwk4fb6dX0joK8EKvUTqK0VWblS2nDsCVMVPDB4uKAIzKXTgv7bvaBzWmrOW1qmtJtFqvu07f1nRHSBH2FA0sBHh8dEgAiEIYBChzLmRRcg/BQp7XhjlJZSTUWw55/v3hbGvp8wHa7H1dti3Pcd10++xgzOIds88jHvKP+z+WaNg7BPijoiME8EkL5QpsXP7N8O/XclI/jqiiWqpTRYXQTr0oB7xpQ4QEpcf6BLg6oo80AgqQUo5dQpBVaLpZXpW8i1axELJq6J/IYNIj/5k9JWh/KWIiQO5gH0PG/t1XZFq/H5lJCw08L9+7BbWlZrt4WrOmDqNSBrGdw/7DkSESh2BDpGOuRk10lpRaFD5KFGFDq0NopLochhoXxmcRbpcWzxfLMtmIMYxPEfZRs1ZnAsO4/jJ8o2rj44R6Yq2KfBPRHIIwIjly7JlEYQ1Gq6gi1/+jUVAXJzyxapUT1kxeXLUqfHIA2swKJr9ajEAwiFB5R80PfhMqhEQhdJBEVi7pLUxaicPSuin43oZyONjanttiYhDmgyyfi4yGuvSfLtb5c22FIWHIFT+hlc1OiPMiVvPqwRHy9PpKqBPF1Rev80HTx4UBez+kU0R2GnhRyBohkRIAJEIA8IuEKHqFuAf6ZQuwCbSfot0VJpo2iPVWj7GSHz3suFML2vw3PYgjlK74fuw94WzL69+cD1OHrfFvean6gxYRMmvh/zATvfT9i8YWP2UT6i9EHfpfftDOhRiMACI1B34IC8fvy4GwUkwW0sVD2x89s7d2ZsQCKAaEA0gk8iIM2hUokEv9Vjt5IIPaqvYUqDwy+nH52dKXJAiRv56Z+Wtk2b3G1JRBzgs7qm1SawIXXhscdks5IJqU9wpvdHE8x7nolIfs6MNMiPt+LzMtd6B3hSu7f4npozJgJEgAgULgIt3ZqKoA1ZMoUOkYqAcNA0UYCZNyQa3HZod6orAiMMgMrCS9RCN1/64IIZT+T7xvlcF+O+H/MBf2FjQh8mcX2E2fs6jGFz8fWmw3XMj8QBkKAQgQVAYPW+fdLb3n4PaRA2FAosVj73nHSq/ea9e2XAa/XopzQgIq5KSYSNGomgy1+5rfchGmFMi6UNkUQIg/auDm+xQRasWJEhDXCxTf8QJpGm0NenrTG6U2kM6bsOVFXdvZ9HRGCBEfjIb/+2nNff0++99FLOI1mUAsmDnCGjIREgAkQgKwKu0CFSEVCvAG9vNCAxEz7qRRfs3qCFarU7AsmCrHAu+EV/oYvBbLGbi95scZ9v7+ujFvRh9r5uPnOJGhM+gxI1ZpQP395/zjC9r8O4JA6C6POcCNxnBJY1NcmWw4fdqLXpvd/qEdFwt0AkKGHgkwiW0tCo+oSGOINEQFpED1MaHJbBH21azyD59NMi9fWgTWdeRs0DpCsg9K0aSSIUIrA4CLxb//t960c/kquoxZGjsNNCjkDRjAgQASIQgUBLp0YX6NuZGdEFIA5AFKRzTlHosP1suyC6oGltk9si3FG9iAj4i924C2Ozj/LhL8bxiEKEoUoAAEAASURBVGGpAeYD16P8hOl9He41P/6YNh6uR4nvJ8wH7jM/vi30Zu/rTYfrJA6AAoUILAACFnGQzbVPGoTZWavHyhdecJfHlSQYURKhTPfBlAZEIkAXTGlgJMJdZNuQhhAgDZI3b4q8/nqqQCLqUKRTGO7exSMicH8R+Mh/+k/yRx/6UKxBW/TvwgvpvxOxbqQxESACRKBEEQgtdAiyIE0UWEqCtVFsP9MuL370xRJFqzgfO2oBHEfv2wIFfyEdtqiPsvf1vo84en88zMUIAByHSZRv34/vI8x+ho7tGMNgpo4I5A8BFEpEygJkFIX50jIbaWB2UXt0bpjwUhqsLgLel2MDiQBmEBuiEVAXQYPxpYopDYqCSLK3N1XbQAsiyltviWzcKPLhD0vbww+76/yx8Aj4xRH90VAcsdS6KuD5b2lHj38YG5OJ0VEXcfC1T33Kh2XWYxRYZMrCrDAJuyrMjhEtiMBSRgCFDju6OtzmihyidkFA2EYxAMgSPPUX7/Z4YTpci6v3F+PmO66ffI3pj+8fh/mfbd6MOPAR5DERWAAEapuaxFIQ4B5EAgT6+QiiEfyUhtc1L3qlFlis1wKLfkoDSAREIyASYYseo9UjegbcKPWUBnRRGNa4RGyTWu0IFf1HkMxIIQKLg0C9/i4+pR09vqvDP7Brl7zzYx+T73/lKzlPhvUOcoaKhkSACJQYAplCh2dbUx0R8M99oNBh84ZmV7Pg0J5DJYbO7I/7vcvfk4QWh37npnfOblwkFjPepKe7NPg6PIYtrnPRmy3ui/NGH/a+f/Pj62Azmz5qTNwbJr5/8+37wD1GJJhtGSMOwqCkjggUJwIgJUaOHHHkRGSrR300RCMkdCv1Vo9JkAaIOHj11VRRxGUat6HEStu73lWcvwBFNmtGHMz+gSEFwQiB2a1TFow8yI4UIw6y48OrRKBQEOgY6pCTN0+mOhtc1gW/vvVoXtksL7wrlb6ZyzxdG0UUOgRJYG0ULRUh7QDRBa7QoRIHlJkIvN73ulwdvCqXB7R9eGWd/NLjvzTTYAme2SI6+Gj50ttiPBf/Cz1mcA44jxoT82bEQRhi1BGBIkXARTdodwYISITrmiKBlAaQCKjvs1yjEfyUBhDu1uoRRIKlNGgQv1SXQEpDm4aGJ0ES6F5OnxZBQbpKpVVIHOhvAKUQELDUgzjkAYslFsInxzkQASIwVwRaTmgqwYCmElzv0DDJtJeVutcvKkgx2P/1/fLCe1+Q5rXhC/3QQoeoiQzCIE0abK7aLPt27BNGF6Txjdj1DvdK160u6RnpkamaKTnfd162NW6LsF4aanu7bk9jC2lfbzrYxNX7b/V9EiHMj6/DWDaurzddtrlEjYl7ghLlGz5IHATR4jkRWCII+CkSVmdhTImD6mPHXKoCSISwlIYHvJQGEPT9SySlIamtFduQnhCQNqQoPPqoJJVJdcTBjRuSVJu2gB1P84/ALm2BeTHEbSnWNwiBIaMCedChBCC2XOV5/e9YIwpzNacdESACRKAgEOjo7JDWkxohEBStY5wp4KQkwPN/+7wcfFzruuxKpRV0jGp0QudJae3Qe0ESaEChizKw6AKPMEBXBLZQDAJ87/m5G+ek/3a/3Bq/JRNT+l2pRCVsIe3rAIst3n296XA9Sh+1oI+yD9P7ulzmEjUm7g1K0DeJgyBCPCcCSxCBGSTCgQPuCXtnS2nQRQrqI4BIKPaUBkcKaNRFErUM1q2TNkQVBGX58lQrRs3h038B5EpDg2xGi8aAvDY1JY/ChpIXBA6w/WVOOKJjwv79+3OyNSOQB+y0YGhwTwSIQDEg0HKyJXqaiD6wugRKDrQeb5WOm0qo6pcVEA73FDr0yAI4JWEQDW3wClIU+kf73YK4pqJGKhJcMgIjfyE9GzHg2+Jes4/S+wt62Fs0gm9vPnA9jt63xb3mJ2pM2IQJfwvCUKGOCCxhBKwoY62X0lDR1CRTShT0R7R69FMaNKhfBjSNAV0aevS4phhSGlD0RiMs5I03RFavFnlnSHEfdFlQskDWrJE2LVAHOaIRCvbH1Sn0x3ZEJlCIwCIggNoFcVIWEKGAGgmW7rAIU+aQRIAIEIFYCDgCINsdU3oRW1o6rihhADLBCAW7oHukI4AsgDDCwMGQ8w/UNBidGJUN9RukrqpOBsbufZGSs7Mlaugvxv3vivnS+4v6MBIBsNq4/phRerPFdd/e14eNCXsTEgeGBPdEoEQRMCIh2KVhwGv16Kc0IPrP79KAugiDuo3t3i1DBUoitGmUQBJvtkEOIBVByQHZutWlIyRBEpw/n7q2aZPIQw+V6G8CH7vQETACIA55YLZ2b6E/I+dHBIhA6SIwK2kQBo1lIKY5fZAFKxIr5MATB0gWhOGVg+5092lHGiTKE7Jq2SqpnqiW6gRiUClRCPgLcdjYYjwXvdniPt/e10ct6KPsw/S+LmqOs41J4gDIUYgAEZiBQJBEQCHiWyAS9A2mTyLgGCRCo26J1lZXXBGRgT2FWBdh8+ZUKgK6KFy9qqESNZJEegLSFlDnYNUqV+tAdu6cgQVPiEAhIWAEgBECuczNbO3eXO6hDREgAkTgfiMQmqZQr7NAer0VSQyZFNoo9mnNHEQXNK1tcluIGVU5IHD82nEZGB+QhuoGaahpkG2rtsnrva9LeRmjLXOAL2PiL9L9xXiY3tfBgdn7etPhuk8i4DwsGsG3j/ITpvd18G1+bEwSB0CFQgSIQCQCIBEglZpjDRlXkmAkIqUBJAKqB2zS3Go/paEQIhHaNMUguWNHiiTo0SQLkAXYtGiiIw0QafD44yyK6D5l/ihkBEAAxC2WCPJgt0YFNRdoVFAh4825EQEicH8QeGbTM3Kp55IbbLBe0wuxVrXUBGQQpmsWOAPvR/P6Zjm0P5WS4Kl5OAcEUAxxfHpcNi7fKHvW73EeEmWJ1AISnwElNgL+YtwW4nASR+/b4l7fjy3qoQ8jEXx734/vI1c9iQOgSSECRCBnBEAkGInw5ic/6Vo94t/yqFaPG5VE2KLXrdVjnx5XLdLipe2JJ0R0S47pqwt0WABpUFsrSGWgEIHFRuCWtgWtH0Yy0Owyl2KJqHfAYomzY0sLIkAEFgeBgVWaR9+oJYkmlTRAqCM2W6xGkAaYKbowsK0ikMguP7j2A+m+1S0jk6hcJVJfWS/r69dLfTXCOjSydOyWIw2qK6ozpIG7wB95Q8BfoMOpLd59velwPa4+jESI8uP7ho2N6+tNZz5IHAAJChEgAnNC4MHPflas1eO4Fh/E8UrtXlAf0eoRkQggEcaUOMB3gRuLlNLQhnoHVs2fpMGcPnvetDAIfKOiQiZGR2c4R3vKp1UfFJAA6JyQq7BYYq5I0Y4IEIH7hUDHSLqN4unWVIFDa6MYiIx/3wPvc50TBqeUVKDERuBC/wWZnJqUiekJGZtM5X1UJapk6s6UO0chxKGxIdd6cXJ6Uk51n5Jd63a5cbpHulP36Bc3HG9r3BZ7fN4QjkDYIt3X4S5bvOeiN1vcF4dEgL3v3/z4Otjc+00EWgoRIAJEIEcEZrR6VOIA0pdDq0ekNOS71eN6faO6XNMlOpWQQL2k8UWKbAAGFCKw0Agg7SBupwXWO1joT4X+iQARyAWBlkst0vpqmiyw6AK70St0CNUzD6RSGHYuS9UgAnlwefyyNCS0E1L6ujvgj0gEtq7aKtjO3Dgj41OpipIoeLhjzQ4533/etV7su93n9lhwohXj6a7Tzt/wxLDcvH3THb/a/arcHL0pTaua3L2RA/JCbAT8Rbot3OEkjt63xb3mxycRoJ9rSgOJA6BHIQJEIC8IWIcGv9Xj9fZ2uaOL+RUaiYDvAlEpDYhGSOh1S2nQ/gdSHWPhv1ZJg7Wax41oxq3qS6sYCP6Zm1Yft5VIsFoNqqIUOAK92ukCb9kpsyNgBQ+NEJj9Dg3r1f9OIHZvLvfQhggQASIwXwRaulpENKKg9Yz+DQJZgIh5tFHEP/5psUKHW6oQn5gSq3tg5yAMjESAbt+OfXaJ+1kQeHzN4/dYoADiwO0BF1VQrt/UjFi4xzCtuJPJH4myoH6+CEQRAL7eSAGMFVfvEwlGIkT58X2TOJjvJ8v7iQARiEQgGI3QqyTCmKYxVB87JjVNTVKhhAIa/NhmXRoQiWApDfhO0T9LSsNqJQ1W62IIlQqw3OzSrVM3JzqGxnNL+cGDklA/lMJB4FRjo1y8fl3KtHClL2Fh+f51Hs9EwAgAIwRmXg0/gy2LJYZjQy0RIAL5RaDlokYXnFKyAP9IIxMLL7yNG06TBgd3H8zUKXjp71/KFEnMZSbopECZHwJ7N+6NdPBPV/5JXul8RVenIjvX7pT3bHlPpC0vLAwC/uLdCANfh1HD9KbDdd/e18chEUgcAEkKESACC47ADBLhwAGpVOIAkQjItBvWhX+ZHmupQlmmG4gEIxFwnC2lYZXeu8ojDbrVHtEGQZlWmzjEweaBAdmsxRNfRueFtGBByzfhhgb3hYTAXDotoD7C0aNHC+kxOBciQASWCAItVzS6QP+Bz0QXgDAAcQCeOE0abK7SNskqaKO47/F97hg/Dv/EYcmVPCBpkIFtwQ7W1q0VFEwEcbCubt2CjUPHuSEQRQCE6X0dvBthEKX3SQTYWzSC2ZM4ACoUIkAE7isCltJg6QPWpWFKyYP+NImAGgg+iYCIxkq97rd6HNA0hDolBBCVgO8hSG/AFiVT6jsOeQA/H67ETChEoPARmEunBZAH7LRQ+J8tZ0gEigEBV+iw66S0otAhOPdgoUOvdgHIAohPGDhF+gfIg/Yz7dJ+tt1XZ45BGDStaYq8P2PIAyKwhBGwBb09YhgxYDrY+Pa56H0iASQCiQNDmnsiQAQWHQEQCVVaKR6CaIQBXejje0ZCjxGBgOgDbPguUqW6at0s4hH7bKSBXqYQgSWPQNxiiey0sOR/JfiARGDBEUChw46uDtf1INNG0Ub1yAKogtEFZha2N1IB5IERBbBzx0xPCINsQXQoqvha72vONzsqLAjEeXMaRgz4OgxkhIGvNx2uR+lBIpA4AEIUIkAECg6BIImAOkq3QCR4JAJiAfCdBFufbrNJ3HSF2fzxOhEoNATmWu8Az2H3FtozcT5EgAgUHgIt3ZqKcCudioCcQyt0mCYKMGMUMcQ2W3RBtqcDeWAEQjY7XltYBJ59+NmFHYDe845AFAEQpvd1mIgRCb4eOhIHef+Y6JAIEIF8I+CnNFx+6aVMl4ZNWmjRiIPpwKBv03Nk4qGzwnXd0Pl57H3vEyYeKBALLEn9XAQb6kPUaKzI6tXSpvUiKNkRqB/WWBqtozFfMQIgbrFEjGv3zncOvJ8IEIGliYBro2iFDkEWWNgfHjdNGqDQYX9/vzSvb+aif2n+GvCpigyBIAFg04+jh+38v6HYyNwTASJABO4DAlsOH3ajjB45IhOf+5yrbYDvKndLGGotBD3HdxkUScTxFt3wx65foxWGdEN0QlWMVo9qTskBgaS23JQzZ0Ru3NAiWFMiK1Yg5k3Zm5uSbNB+2zt3ijzxhLR5BSdzcFtSJj8zOSkve0881xobRgDEJQ/YacEDn4dEgAg4BFo6NbpAec1shQ5BFkAO7TnkOiI07Wly5/xBBIhAYSHgkwWYWVR0gc3atydxYKhwTwSIQFEhgJaOIAdAGujSdIYM6iJ1Wt94+3URUBuhRnXLvVaPuO/GLK0eZzjmSXYETp0SeeONFGGwcqWLNJAxjWHt1eoTXdokUxfFjkh47LHsfkr86lzJgiBsIA/iEAe4v0XTgVgsMYgkz4lA6SFwT6FDS0VId0TwowtAFvjCTgc+GjwmAoWNgE8MhJEImL3pSRwU9mfJ2REBIhCBwOgzz8jUpUtSi5D4tPRv3iy39S33bSUOQBJAVuhb8Do9Dmv1iGiEbK0enQP+yAmB5Kj22rquSSHoQvHBD0rbAw9k7ktqxIGcPSvygx+kdLMQB2x5mYFu3gcgAdA5IVdhscRckaIdEViaCLhCh91eoUO0UTRJkwZ+dIFd4p4IEIHiRyCMRMBTmZ7EQfF/xnwCIlCSCKDuQSdC31VAEoAs8MXOb3s2IBEQobBc7a3VIyIR8CLFWj0m9Pi2+u7Wt7Xo0lDNlAZFIQdB+kFtrciaNTNIA9zZptEHSRAJ3/62SGenJJVcQLrCrr4+uRhwna+37QG3JXvarL+/cTstWJSCpTuULHh8cCJQIgi09GgqwpCmIpxt1X8A9aEtuiBdswAwNG9odjULgtEFJQIRH5MIlBwCRhbYg7M4oiHBPREgAkWHQG1TU2bORhJkFCEHdQcOSKXe09veLmNKHKxUEgExCRVa8wDkgW1Ib0CrxxtvvilP/tEfyZguvDTAXvrTKQ2vae7+ownQC5QZCNTXu/oFgn2YVANhFaQrYEvLAdObgvu8I2AEgBECuQxgtnZvLvfQhggQgeJCwBU6PK1kAf4kgzDAFkhF2Fy1Wf7tB/6tIw6K6+k4WyJABPKJAIgERhzkE1H6IgJE4L4isHrfPkcEzDYo7LBBag8fdnv7cUdJAnSSGtbc7jI9tpQGrQMl/+XllzN5Xc5e35LvPX1alitxMMRIBIPQ7dvKyyW5a5e0zdB6JyiYiA4Lq1ZJGwomUu4rAkYAGCGQy+CwZbHEXJCiDREoHgRCCx2iLREIgzRpALIAgjaKbIXooOAPIkAEFAESB/w1IAJEoGgRWKYRBLOJTxqE2fqtHnF9SsmDfiURtD11qCAaYdNXviJ1enVAyQOkNPTocQ2JhEjSIHn8uMjrrzvSQB55JBRXKhceAZAHqGGALVdBfYSjR4/mak47IkAEChSBlost0tqh0QUgCcCMI8rAogs8wgBkAYSEgYOBP4gAEfAQKNMvBHz14wHCQyJABIoPgZFLlzKRB6N6bDIbaWB2YfsrWmTxeE/PzIgDNdylpMFTuiHwHtEJKLBodREGlTwY272b0QiKCSRZpQj9+Mcir7wiguKJIHq05kRboB6FM+aP+4bA/v37Y42FOglLqdNCS8tq7TZxVTFIpRw1N5fp8/E9SqxfChoXBQItV1qk41pEocP0EzC6oCg+Sk6SCBQEAvyXsiA+Bk6CCBCB+SCAegd+CgKIBIhfB8Ep8vADNaM6dUMtBJAH2KwuQqO+yU1oeDeKK6KmFKIRSjWlIXnhgggiDQYGXMFE0TQG2btX2sbRRJOymAiw08Jios+xicDCIpBLoUPMAIQBowsW9rOgdyKw1BAgcbDUPlE+DxEgAgtCGBis4+97n1zTHH3URrBWj4g6sC4NRiLgXeYmDfP2UxpKhURIXtReCW+8IaJFKF2nhW3bNFRD6x+QNLBfo9D9rbo6qR9GDPFM+a4Wk3y6In//XLPTwkx8eUYElgICmUKHU/o0VujQHgxMtgraKL7y+ity4IkDTEVIQcKfRIAIxEAgf99EYgxKUyJABIhAsSKAugpbsBBWQWTDde3SsEwXyJXHjrl00ahWjxuVRNii91irxz49rlpidRGSY1pm8swZEUR8aMtFQUFKxcoVQ2RBRP3EZ5dvKEEwgbQOT1aXpROQPd18D+daLBHj2r3znQPvJwJEYH4ItHS1uHoFrWdaU2SBtVEEc52WYBtFtlM0ZLgnAkQgLgIkDuIiRnsiQASIQBoBP0ViRNs9otXjuJIIIBTQ7rFej/2UBqQ1uOKKaRIBrR5RZObGUklpOHVKRNtYClosvu1t0rZ1qwgJg4L978UIAHZaKNiPiBMjAqEIuEKHp5QsQHQByIIJ3YxfTJMGiC4odZLgfN95GZoYkompCU0fLJfKikqpqaiR7Y3bQ3GlkggQgewIkDjIjg+vEgEiQARyQmAGiYA37ip9R444EsFSGrRU4D0pDUhzeECJhFrdW5eGYkxpSPb3pzongCjQSIO21av1ibLLqcZG2dWH2IuZ0qs+FuIt+8xReAYEQB7EIQ5wT4t2HVlKxRLxTBQisNAIdAx2yMn+k26x39HbIS889UKsIVHoEL2DW19NRxcgHcHKm6dJA5AFkFInDIDBdy5/R7pudcntydvKqZRJbVWtK3aM8zd635C3rXqb7FizA6YUIkAEckSAxEGOQNGMCBABIgAEHk14MaARkFhRxtrnnnMWltKwWYsDotUj0k0rtEaC1UVAJAJeGlWqDnURMIKlNPTqcXUxpDQg0uCtt7RQvc7+xg1Jrl3rjtsm8CosWo5oekNZORC5K/nM57/rlUdRCLBYYhQy1BOB+SHQclUX+9qvt/VierFfr/7wzVtbIj7/3efl0PZD0rymOXKQjhElGzpPpsgC/Cm1Nooz/2S62gUkC+7CiEiDnuEeGbg9IFUVVbKscpnUVtTKxPSE3J64LVNTU9I70ivn+8/LtlXb7t7IIyJABLIiQOIgKzy8SASIQKkisFm7AWxGO8E8iB+NUPVC6i0TiisOKYmg5QMlocd+SoMVWASJgLoISGlAy+3+Ak1pSKJzwpUr2kai25EG0qt0B1ouam5+Mpifv0Pf8PziL0qbfnGjFAYCLJZYGJ8DZ7G0EHj+6PPS0dkx86EQYLVcN13449rz156XF37iBWleN5M8cIUOUbcAZEFEoUN0Rdi3Yx+jC2Yi7M6QnjA9PS0N1Q2yrXGb7Nmwx+kv9F+Qa0PXZGh8SIbGhqSyvJLEQQh+VBGBKARIHEQhQz0RIAJEYAERKFMyoMIjETQCVYaVSChTEiEspQFRCZbSMKj3dhUSiVCvr9E2bRJ54gmRrq4UakHCYAGxpOv5IzDXegcY2e6d/yzogQgsDQRACtxDGtijDekBctQQWobIg28+Lwef0BSDjXp+S6MT/EKH4Fe96IKGRINgYxtFxSWLlJeVy/Lq5VJdWZ0hDWC+ddVWGRwflNHJURkeH3b7LG54iQgQgQACJA4CgPCUCBABInC/EQCJAKlMEwlTSh4gpQEkQrDVo5/SUKf3WF0EjYaVmrQf+Lqf0ob0BE3DcFsuAzPaIBeU7ruNEQBxah7Advfu3YKoBQoRIAIpBJBekFUQSYAwsnSNgtYTGl1wUc/xB35cNyt0mCYNEF2we4P+d7a+mW0UFZ7ZBBEc58rPSVXi3qhBpCqMTWqKnJLbFeVcBs2GJa8TAR8B/hfjo8FjIkAEiEABIAAiwU9puP6Zz7iXTmj16Kc0IN3V79KAugiIRhjThVwxFlgsAOhLfgogDzqUsMKWqzyvKTVHjx7N1Zx2RGDJI9DRlcN/P1bY0NDQ+rLuD32aNABZAEF0QdPaJrc5BX/khMBjax67x+7V7lddbQNEHKysXin11Sg6QSECRCBXBEgc5IoU7YgAESACi4CATyL0prs0oNVjXYBEsLoIjbrgS+hbYJAIeFnVXUgpDYuAXzENWT+sVFDF4v+zjGKJ+/fvjwUdyAN2WogFGY2XKAJZ0xSyPbMXXcBUhGxAxb92qvuUdA51yo2RG65gIgiDlStXyp71qdoH8T3yDiJQmggs/jeU0sSdT00EiAARyBkBv0sDOjRArrW3CwosWqvHYEoDIhGQQosCi35Kw3wjEdZrCsVy9d2phAQiaseVoKDkD4GfmdT45Up8mosr7LSwuPhz9OJF4FLPJVeHYHBqMPUQK3WHKAJEGGgNA5eioLugoHbB7+z9HaYiBIHJw/nUnSkZmxqT8alxmdL/oXDixFT2jj95GJYuiMCSQ6BMwwuDwVJL7iH5QESACBCBpYoAiIReJRGWaQRC5bFj7vspUhr8Vo/V+vBYioIpxmatHhEZWxlj4d+kJES9kgb4Dozt9fT+jvoYVSLBajWoOic51dgoF69fD23HuJrFFXPCcCGNWpQkilPvAHM5ePBgwRdLbGlZrc91VWcLak20PkOZRkvwPYoDgz/ygsBv/N1vyODNQRmsUPIAXRFQuwA1DbxCh2EDHf0EU37CcAnqfnDtB9J9q1tGJgGsSH1lvayvXy97N+4NmmbOz94469IUuoe7XVHEykSlrK9bL+9/6P0ZGx4QASKQHYGE5jN+OrsJrxIBIkAEiEChIlCp4ZYr9uyR2ne9S6bf+U7p1xaSEw89JJ26T2zcKCPa5WBMJ4/N6nElVLfym9+Ujbo1KBGA46kNG2R8/frIx9yipEGt2oJpxvZj3fDyDH7H1d+0+oCUa32FXKV72TK5eeuWK1Ll37MHxRYpi44Aih5C4tQ7MFu7d9EfImQCHR21+kwobZ9axa1fr21Dk7Os6EL8UEUEggigjeLvHfs9GRwZlLER/euIXzP84cUfzXTtguA9/jkKIGIBTIlGAC0V0UrRdUeYGJWp6SmpSFRIfVW9iyJYWYMQj3tlbe1amb4zLeXl5TI5PSl9I33u/PG1j99rTA0RIAKhCJBiD4WFSiJABIhA8SGAlIbaw4fdxBGJgO+rlaob0wV/tlaPiEawVo/WpcFPaUC6Q7X6wAszfPd9UzcUZgzKtNZWUDY6qI4839XXJ7uqEQ8h8vLEhHy4AEL0IydbohfYaaFEP3g+ds4ItHS2uD+ImTaKo3qrtVHEH0wVFDpEgcNnNj0j7WfbBWkMl8cvu2s4tjaL2FOyI4CWitjO3DjjUg9gXZ2olh1rdmRuvHTzkjStbMqc2wF02L5/7fvSdatLRiZG5HzfednWuM1MuCcCRCALAkxVyAIOLxEBIlC6CFxZsUI2DwzcA8Br2krw0SJ+I466CBMago73qxV6bCkNaFqFAouW0pDQY6Q0IKIgoXawR8Rt6quuHkRIhRbWi5uyEOGK6gJCIG6xREy9UDstMFWhgH6xinQqHSMdcrLrpLSebk1FFFgqQpoosMcCYbClaoudhu6NOACxcPgnDofaUJk7Am/0vSHDk8OujkFdVZ1sb9x+z80nuk7IiesnXDvG92x5j2xbReLgHpCoIAIhCDDiIAQUqogAESACQODI+Pg9YfTbNcyxmMXv0gAS4dbJkzKoe5ADfqtH69KACFtELuD7MKJuZ5Np9ZeIUTdhNn+8XhgIxC2WiFmjRoJFLBTGU3AWRGB+CLS82SJotYjOCY5JRXSBSZo0OLj7oNNMDU4JCiXOJhZlsG/HvtlMeT0HBMrLymVsckx6R3ulbiycOBjVFAdEKdRW1ZI0yAFTmhABQ4DEgSHBPREgAkSgxBAAiYBFPqILQCIguiCY0oBIBFAl2PzvyHoaKnHTFUKdUFlwCDTr7wkKH8Yplmi2JA8K7uPkhGIg0NLT4ljT1rMaXZCl0CGiC1786IsZzyAN2vV/uZAHuAkRB5T5I4A0hpujN2V0fNRtKKTYWNPo0hGQltB3u09AHCyvXi5R9RDmPwt6IAJLEwESB0vzc+VTEQEiQARiIWDpBZWaagC59MlPyurLl2Wddmgw4gDEgkmtHmzVDa0eb+p2XbfLDQ1SowtMpDtQlh4CRgAYIZDLE5qt3ZvLPbQhAoWAAAodulQEFHcBYeD/AcQfRRWQBSsSK+TAEwfuaaPoUg/WHpb2M0ofnG139lE/GG0Qhczc9A01DVJxq0J6h3tlcGzQpSQc7zyuJHnCtWOsqaiRNbVrZF3turkNwLuIQIkiQOKgRD94PjYRIAJEIBsCD372s5LQoogTGomACNzpgDH+8ejTDUUSEZXwoG5vU5Lhtm6DGqI+phX5/QKLepmSJwSSQ5o0omk0batX58lj7m6MADBCIJc7YYsuC4haoBCBQkagpUujC7RdzIzoAvzx82oXgCyAHNp9yEUJzBYpsO/xfc7eJw9wT9OaJqe36+6EP/KCwMOND0vPcI/ruIACiOi8gC2h9YlqEjUu0gCkAYsi5gVuOikhBEgclNCHzUclAkSACMRBALUNrDg4jn3p0uiCW0oShNVFaFSyoVwXi2O6UMSLuW7ttEASwUcv/PhWXZ2cVULgKe0w4ct3Jyfl6YrUP9dJFOb8xjf07eeYJJ99dtHIA7RdtNaL/lyjjp9XEqpQiyVGzZn60kGg5aJGF3S0phhSFDq0wi6AIE0agDAAWQCJu9iHvREMtneO+GPBEEDRQ8jrva/L5J1JqSircJEHSGWgEAEiMDcESBzMDTfeRQSIABFY8giMPvOMTGlbx1olCFAk8baSBaO63dyyJfPsNXpthaY01OkeKQrWpcGKK6J+Ato5IqUhrNVjxhEPHAK9d+641pQ+HKvLUiuXpJIFcvy4yA9/qGEeGufx1FMiixB1gLmhWGLcTgsgD3AfhQgUAgItV7TQ4bXshQ796IK4ZEHwGUkYBBG5P+fbV2+/PwNxFCJQAgiQOCiBD5mPSASIABGYCwIgCnp37nS3giDAeVCgu+3ZgETAMne52huJUK3neIlXqW+pN+riEbQDWj0iEqEfeoawKwrR0nj7tiQRZaDYuu3qVS0woW/NlGRYTInbaQERCuy0sJifGMfOFDoMtlFM1ywwhNAZwdoozpcwMJ/cEwEiQASKHQESB8X+CXL+RIAIEIEFQmD1vn3S297uvIeRBsFh1zz3nFPhnnElDnD/kNY7CLZ6BJFQpYvIE42N8uHf+A2X0oAl8I10SgPeuttb9uAYJXkOggAbWoEiZaFSYzuQspCORFgsTNhpYbGQ57hxEXCFDl9tFbFChyh2aJImDayN4qE9qXQEu8w9ESACRIAIpBAgccDfBCJABIhACAKbBwZEg8LvkUexYCsh8cmDbI8Nu9qmJmdSe/iw248gzSEdmj6mREGw1SPSiP/LkSNShgWxieb3f+TP/kyW7dnDughpTPqWLZO2X/kVSeL89GmRr389RRwUwO/iXIsl4lHsXhxTiEC+EXCFDrV6a+sZJQxAFCDsCcSB9ye8eUOzNK9vFpIF+Uaf/ogAEViKCJA4WIqfKp+JCBCBvCBwAHnkFFmmhMCokgBRAtIAW1CMSIAe7R6t1eOUkgj9GokwswTg3bsRobDpv/939/3eUhp69XJ1iac0tCkGySefFPnud7WvvHZW8AmXu/Dd9yMjANhp4b5DzwFDEHCFDk8pWTClF4OFDtOkAaILSBaEgEcVESACRCALAiQOsoDDS0SACBCBUkfAkQIKAqIHQB7Y3nCJIg3setgeJEKVRiKUaaqCXL9+jwnqHizXzQosorgi6iKgSwPWAn0l3KWhTSMyXGcFpCkscqqCfhQZAXnATgsZOHhwnxFAoUPR2qGZ6IJRnQD+WCCYCUVXVJiKkMKBP4kAESACc0WAxMFckeN9RIAIEIESQgDRA9hWp58ZBALEjypwijz8QOvGzlOnBPEeVmDRSAToHlAioVb3g2rXVYokQgERBv7HDfIAnRPiCIslxkGLtj4CHSMdcrLrpLRaoUNNS3CpCJb5lN6jM8KLH33Rv5XHRIAIEAEiMAcESBzMATTeQgSIABEodQQWgjAwTCt0AXp9elruaMoCujSgHWSw1aN1aQi2euxRJzVKKFDuPwIslnj/MS/FEV2hQ9QtQK4TaheEFDrMZxvFUsSYz0wEiAARCEOAxEEYKtQRASJABIjAoiKwxSuw2NneLsuUPKg4dsxFHqPVIyIQqtMbXjSiS4OlNKAuAqIRxnbvLqoCi/XD+iTomhCQp0N0AZOCOZ1rvQM8gN1bMA/DiRQMAi3dmopwK5CKgEKHFl2ghw2JBrcd2n3IzZttFAvm4+NEiAARWCII3PsNZYk8GB+DCBABIkAEih8BlyJhJMKBA+6BerUTA1IlVmo0Ql2ARLCUhkYlEspbW11dBKwtuoskpeFnJnU1hHaLRSxGALBYYhF/iAUydVfoEKkIqFeAMKNx3dI1C4w0QHTB7g27XXcEkgUF8sFxGkSACCxJBEgcLMmPlQ9FBIgAEVh6CFh6RO1zzzniAE94TaMRLKUBJEIwpQGRCAm1C6Y0oI4CZeEQAHnAYokLh+9S9tzSqdEFGnyTrdChn4rQtLZJsFGIABEgAkRgYREgcbCw+NI7ESACRIAIxERgdQ7F/zIkgkUjaATCdSURNu/dKze11SOiDNDW0YorIq3B6iJsTHdpsFaP6OJQSSIh5qc0u/kL2jlj//79sxt6FiiuiPsopYVAroUO0RlhS9UWBw6jC0rrd4RPSwSIwOIjQOJg8T8DzoAIEIEiQuC1qSl5NIF32JT5IrCrr092VWNJP3/xUxrQ6hGCSAR0ZRtSIgEkgl8XAaP6dRHQ6vGO6m4USUqDTrUoBCRAnE4LiFJgp4Wi+GjzMkkUOuzo6pCOzg7XTtH9B2ue0/UL2EbRAOGeCBABIrC4CJA4WFz8OToRIAIFisCVFSvkeE+PHKhCA0CRlydQwltke7lXjctp+KNQEShLRxFUpomEMV2UDiuJUKZ7fKoWjWB1EfAPorV6HNB7i6UuQqHij3mx00Ihfzpzn1vHgC70p/XzXRU/5SdT6PBsa6ojAkKBAoUOmzc0u5oFh/akCh3Ofaa8kwgQASJABPKFAImDfCFJP0SACCxJBIwwWJIPV2IPBSLBSIQpJQ/60ykNFXrsRyP4XRoQW2IpDb16XJ0mIxYVuscf1wrzWmJ+06ZFnUaug8+1WCL82725jkW7hUGgY6hDTt48KR3nNTpgXEkDhOekiYMX3p1baolro4hChyAJrI1ioNAhogtcoUMlDigzETjTc0ZuT96W6TvTUpGokNqKWtm+evtMI54RASJABBYQgbKjR4/izz+FCBABIkAEPAQs4qAskG+PiAOmKnhALYFDpDRMnzwpw9qtoT6sS4M+I1h2kAhIaUCB9z6mNCgK8QQpCHE6LcA7Uh0QtZBPaWlZrfO4qi5TKUfNzWU6Dt+jhGGMFIKW1zSd4JKSBb406AngA3nQ2CyHdhyS5rX3fk6hhQ71nkxnBD1kKoKCkEWOXz8uPSM9MjIxInem70h1RbXyNncciYDjdXXrZH3demla2ZTFCy8RASJABOaPAP+lnD+G9EAEiAARIAJFjAAiERK6VTzzjPSlWz2uSLd6DEtpgM5SGgb1vi6SCDl9+uy0kBNMBWXUclJJA9QfCMqgKvAfgn6L7LjeIc9f1aKWP6kkz/pmyRQ67Gh1xAI6JLgoA4suSO/RGeHFj74Y9MzzAAJdw13SN9onNYkaqamokdqqWpmcnpTRyVEZHh+WzjudLgKExEEAOJ4SASKQdwRIHOQdUjokAkSACBCBYkTAFVjUVo+QkUupLg3W6rE2pNWjdWlAq0f8Y2opDT16XDOHt+S36upk/fe+JxtQh0F9gJQY2L3bRTno6ZIQkAdxiiXioVkscfE++lDSwKYzrgdIO8Avq4bhPP+/npfmx5U4ANGAVARUJjXxyAKoDu0+JOyKYOBE78/dOCe3xm9JoiwhWxu3ypMbnswYv9r9qotEuDp0VTr1fxQiQASIwEIjQOJgoRGmfyJABIgAESg6BPwuDSAROtvbZZmSBxXHjrlWj8sDKQ3BLg0gEbDwH9OF/1COJMI6JQ3+P20nObFjh/z+hz4kqMPw4j/7Z9J4/Li868/+TBK66LaCj0UHaHrCLJZYPJ9cVtLAHgNpByZKDnScU9IAuTyBGrKILgBZACFh4GDI6QfqGVSWV7p6Bj5pgJt3rtspx68dl4tTFx25cOnmJaYr5IQqjYgAEZgrAiQO5ooc7yMCRIAIEIGSQGAGiXDggHvm3nRKw8p0SoNfXBHHaPXYqFt5a6uLGMA6KluXhmVqu0UjF355zx55U4sf/vXHPiYndTNB5MOkXi8/eNARCKYvxr0VPIxT78Bs7d5ifO5imzPSFGILqmalSQOQBSsSK+TAEwdIFsQGMnVDVaLK1S+oTFSGekhUpOp03LlzR6bugLGhEAEiQAQWDgESBwuHLT0TASJABIjAHBDo1S/BqwNFKefgZkFuAYkAqdWUBkQiQBKqG9eF/Ui61SO+4gdbPeLrPVIa6nQfbPVYo/du1GuI+t5w4oRU6IZ0B5840NMlJUYAGCGQy8PBdrdGcOS7WGIuY5eizT0RBxsUBdQrQBpCqjttKCwWXdC0tkmwUeaOwMOND2smSJlsa9wW6mRkfMRFJKBI4rZV4TahN1JJBIgAEZgDAiQO5gAabyECRIAIEIH5I3CqsVF29fXNcIT2l09XFMc/TUYi4AH8Vo/TSgTcTLd6TOixkQhIZ7C6CCAKtug5UhqGdG9t7LEmA2kQJdO6eEbKwlIQkAcdig+2XAX1EbQbVK7mtJsHAj/34M/JpZ5L8mrVq6koAvzyom4BSIN0zYIw9/t27GOEQRgwc9RFkQb/8NY/yI2RG67LwpraNXP0ztuIABEgArkjUBzfznJ/HloSASJABPKCwOaBATke4omtGENAmYfqyNiYlGmLy6UkIBGqtI0gBCkGWGsNKZEAEsFPabC6CHh61JnDPhtpoJedTMHXEiEP0G5x//799mg57UEe4D7KwiHQcqVFjo0ek8EabZ8QUehw4Uan52wIoJZB70iv3Lx905mtrV3rWjJmu4fXiAARIAL5QIDEQT5QpA8iQASWJAIHqtBvjEIE5o6AFTOsTC90x5Q8GE6nNOC3q1Y3pDaANMBLXKzRfGlYv14a3/1ueeuxx2TNH/yBjN68KWt9gyVwDBIgTqcFRCiw00L+P/iWnhZluERaT7emQmCsjWKA1zu4+6D09/fLsavH3CQGp5Rc8OTSjUveGQ+zIfCDaz+Q7lvdMjKJcA6R+sp6WV+/XvZu3Bt6G+w7b3XK7cnbrmDi+rr1snH5RhZFDEWLSiJABPKNAImDfCNKf0SACBABIkAEIhDwUxrQNeFSb6+s0I4Njf/4j64Yvd02+I53yNpt26Rc0zaGNfpl5MknZfB//A8p1+iM4dFR2Xb+vBRyLQh7jlz2hdZpIak1NnRlrCyO0jiIhlm+XNq0VeZSlZZLLdL6aposAHPls1dp0gB1CyBbqrbIef3dg+xcttPtjTgYmBpw1617grvIH5EIXOi/IJNTkzIxPSFjk2PODsUQo4oc/vDaD6V/tN+RBui00LisUdbVryNpEIkwLxABIpBvBEgc5BtR+iMCRIAIEAEikAWBW0oIjAwPy+VHH5Vx3Y/94i/K/ne+c0a9uempKRddUF5ZKQnblETY8MMfyr7//J8dybBc0xVybfWYZToFcWmuxRIxebt3vg+SHNG3vmfOiKDopZIz0tAgMq39BrUOR3LjRpEPflDalkgUUkuXRhdoREHrGSUMQBTghTcKbSR0S0vzhmZpSDTI1ED2av2wgWDPgohp8HLYbV21VbCduXFGxqeQrCRSnaiWHWt2zLj79d7XpXu426UmJMoSjigAacBiiDNg4gkRIAL3AQESB/cBZA5BBIgAESACpY3A5KpV0l9bK2/pm+vJ8XEZ1wXp+JUrMjo4KFNaEPJ//tVfyc9+6EMZkG7s2CHDmpZQoQvVhBIGRh68W1s06lJWkObwgOb6I9VhUGsqdCmJ0Pv+90sV3pQXqRgBsGidFk6eFHn9dc0Z0aQR/axk5cpU1IF+TnLtmshrr0ly715p08+rWKXlokYXnFKyAFwAyAK/0GGaNEAqwqE9qQKcKI740t+/lPPjojAiJR4Cj695POsNXcNdjjSoKKuQlctWylObnspqz4tEgAgQgYVCgMTBQiFLv0SACBABIlDyCAxqdEGDhnZX6IL+x8uWyVhXl4zrm+0xjTQoS7ecLE+kVmwgBPo3p0LCExcvyuYvfcnhd+OZZ2SkqUkSSiLo8jXTpQGFFrH2q9SUB7R6fESP/1TflpepP6Q01Otb8w06Xt2tW3qlOGSxOi24aINz50S0WKc8+6y0PfRQBrAkjq9eFXnrLX0jr5/V49kXepkbC+QAhQ5FHysTXYBqnSAOkIaAwhoqlorw4kdfTCnSPxFB8Olf+rQjD0AiUO4vAqe7T7vOCRi1qbFJdq3bdX8nwNGIABEgAh4CJA48MHhIBIgAESACRCCfCIA0MHla31pbFwkjDexa7ZtvysX3vMdO3R46yIPp/YguYG/83M9JxbFjbs23XKMV/C4NqGU3pqQEIhRARtzUbXDLFkckLFc90h82adh93RAaQBaugDyIUywRTzLvTgs9PSIgWFDPwCMNHEoa0SGKnXz/+ynQioA46BjpkJNdJ1OFDhFVECx06NUusJoE+x7fl3q+kJ+Hf+KwtJ9pl/az7fdcdekJa5rYgvEeZOavQOeEgdsDUl5WLiMTI4K6CDhuWtk0f+f0QASIABGIiQCJg5iA0ZwIEAEiQASIQBgCI5oTX6uL+WwSbD25Rd9i12qXhVwERELt4cMiBw44894jR2REIww2IsJAow5AIqDrgqU1YI/iio5E0H2ZRixs+M53Cr4uwlyKJc6n00IjIj9AGjQ2iqxbd89H0aapJckVK1I2iDgoYEGhw46uDuno7EjVLshS6BCEQTayIPiYZouuCU1KFLCeQRCh/J6f6TkjQ2NDMjwxLKMTo444qK6o1iCRdJiINxxaMu5Yu8PVTPDUPCQCRIAI5BUBEgd5hZPOiAARIAJEIFcEdunb74shxqvTIfwhlwpO5Qod6tv8S5oTP6Vh7uO6QH+/PleUfFgX86/h7bXKo1iEaiqDZtXnJJuVNKhtasrY1j73nCMOKtO6v1DSoixdFwGEgU8g4PjjTzzhat/VqYcBfYveXcDFFeda7wDg2L0ZoGY7mNSqgCAO6utT5EGYPa6hy4J2XEgqAQMyoVCkpbtF559ORdCUhEyhw3RUAeaJwoXYcokuyPZcRh5ks+G1/CBQmaiU5dXLZV3dOhkeR8gIhQgQASKwuAiQOFhc/Dk6ESACRYYAFn1uwVdk8y7U6R6ori7UqWWdV88DD7hChyhsOKa/E+NaS+C2LtxR+PBlJQRAEERJ8Pdn9b59jgAY1eiBKAmSBmbnEwkf1oiHv9YIBtASE1rYr0zJDEQbVCjG6M6gwfgCtK0uwkati7BFz2+nSQSUVXzrIx8pmLoIRgDELZaojxGPPABxoDUnBJ8ZiiKGCdoxws5IBiUPFltcG0W/0CG4DHsZ7aUi7N6wW5rXN8eKLljsZ+P4Io80PuI2YkEEiAARKBQESBwUyifBeRABIlBQCFzR0OTjmvcczEXfjr7ulJJEwAod4uF/rNt4d7crcohihyZW6NDOc9mDOFithkg7gPS2t7u9EQlRpIEzCvx495o18uPeXg0pGJC+zk65U1MjYwiz1wXxDSUIEumUBhAItlWliyt+W+sxoKjigNZFwH45WhKqbND/DharLgLIA6QhYMtVQDQY6ZDTPRpF4AgB/LcdlYoAosDsQB4sEnHQ0qnRBcpxZCt0iK4IkC1VW5hOkNMvAI2IABEgAkQgFwRIHOSCEm2IABEgAkSg5BGYUehQq+z/pS6skRIQlDKEtGeJOAja27lFD7g6BqaMuV+tBAE2SG+a0ACR0KcREZUvvOD0Y7oIH9a6CmW6x3vzZbqBREDEhEtv0GfCc/XrIhpECAosNugzoT7D1jfekBsaYbEmaoHtRsjvjxd03vv374/lNFaxRKTG4HlACGibzFCBDTaQC/eZNAgtdIiwkkB0gd9GMfQZqCQCRIAIEAEiMA8E7v3GMw9nvJUIEAEiQASIQKkgECQNyrTlIkgDRxxodf7FFiMQbG/zKdPIAyMRLn3yk7Li8mX5zle/KhXpwopIazACAXtHIuiC+Vc1jQF1EZBGL+pjvnLnTlIUMlc6AOtxQFZX1xbqFuRBnE4LsYolgjTQVpkuXQGkT5iAUAAZBNIA6TXpOhVhpvnSZQodarFDwbRSASAp92nSwKILDu05lK9h6YcIEAEiQASIQCgCJA5CYaGSCBABIkAESg0BK3T4puazT2qhw/fogjqbIG3FpQWoUbkuuh9ZrQkHeu8jDz6Y7baCuvbgZz/rUiQe/va3ped97xOUdQRRUKELZCMPQCT883e8w817U7ouwpgSB3jp3TeHAosjI0k5c0bk0iVdC+tiWEszuBf9qCm5cWNSPvhBrM9nEghz6bRgtRFmTVsAIaA1IUTbZUpUq0oUQwS5oLi0LSBpkCl0eLY1RRYgC0YDIVz/Td1Bmjc0u5oFJAtSePAnESACRIAI3B8ESBzcH5w5ChEgAkSACBQoAih0eFlfd4MsGNc3y2NaGwBh+3+phQ9/PkvxxvOoCaDPBMLgke3bC/TpZp8WUiS2Y1NT1FkYfeghOadpDqPa/nF461ZHIuiSOpPSgLaPqItQpfsHlEhAYsSfqi3SGqq1pkK11ldYr6kRUXURTp4Uef31VOQ/siqwZseL/itXUmv3114T2bs3KRMTM8kDIwCMENBhZxWztXvDbuhD7QKwF0jtQH2IMNE6D6LP5iYbdn2eOlfo8LSSBSAJEF2ALSQVwRU6VOKAQgSIABEgAkTgfiNA4uB+I87xiAARIAJEYNERGNI2iMt14f8jXfCPaUX9cSULxnThOI7q+mlBgcCXlTyI6pDgCANEGSwhAYkAIuC9eAv/8MPSq4vq3m9+Uy7t3OlSGmqVUNErM0gEvBRHgUhEKqCrxKgSCFZgsV5DCjag44S+yUebTUQbnDsnohyNPPusyEMP3SUHHnooKVo6QrQxhCs58Pjj9wJrBIARAvda3KuB7e7d2lkgW3oF8iQ0agRtGZMaUdCG9IW0JG/cENHaDo44WLvW1PPehxY6RIkFEAZp0mBz1WY3zosffXHe49EBESACRIAIEIH5IEDiYD7o8V4iQASIABEoSgRAGkDerq++/0oXimirGNYRAYvdKHGpCVEXl4gez7/6Ax8Q0Q3RCJ3t7bJMyYMGTeNAtAW6NPzvY8ckoVEGfmpDQkmEMl1839QNxRV/4s//XIZ04Y4X97o2d/UMfNIAcGFdjyyA738/BV4YcYArIA/idlpAfYSjR4+mHIf9BAHU2JgiD86elSQiSNCCERNGiASKMWzaJDLPyJKO0Q452XlSWjs0ugAkAXgqRBnYr5lHGBzafcjNdN/j+9yeP4gAESACRIAILCYCJA4WE32OTQSIABEgAvcggLfc2Rbs99wwT8WHdHH7cqCaPuoXrNHtfs5jno+x4Le7aITDh2eMc0eJg9saUVCmi+yEVxfBkQgatYD9JzRaASTDj791x5EGWJ+vWzfDjTsZH2+TFSuSzsZ74X+voWpAHsQplggn2TottGlHheTevSKnTon83d+JfOtbqQgD/F6g7oGSH7Jrl7Rpu8u5SMuVFum4pm0lO6MLHVp0AQgDkgVzQZn3EAEiQASIwEIiQOJgIdGlbyJABIgAEYhE4JSuIC9ev+7a/CEdAGkBkKf1bXU+pHvjRnkrXejw3bMUOsR4IAsgj862anVW/AEE0KHhQ7r/a80vmNQaABO6gURAcUWkLoA40Hf2groIeGmPaIP6+tTL/W0aBYAuDX3vf7+8pb8LSGnANdQ7QNmBqqqkgEwIk7kUS7ROCyL/LsylaD6DPpC+8j9xQqS7W7TIgiuGqBUbtZDFI7FJg5aeFiUdRFqzFDrEREAYMLog/COhlggQASJABAoHgfx8Oyuc5+FMiAARIAJ5QWCzhl4fD/HERWUIKHlQGWkwX1eu0KEWunOFDlG7QBejo/pZHtFF4IEshQ6j6hjMdz6lcv+79U08Okz0aZ2AO0oeaMaBTGjVwzKtftij5AJSGvRjcB0PlUsQFEVEtMJGJQ++feGCI4+Q0vCEkgv68t9tIBmUf4iUudY7EEHUwAfu8dsGomDHDrehzoErxKDkR1u2SdzjRSRT6BAgWKFDs0txU44sWJFYIQeeOMDoAsOGeyJABIgAEShoBEgcFPTHw8kRASKwmAgciLlgWMy5lvLYMwodosChFjpEkUMUPTQpS0cT2Dn3+UVgtTIB2CC9+hm4NpXKFPTpZ1HxwguOJMC6HKQAPgoEdSASoVo3dLCw1Ab8J4doAyMPLj+2FS5lm5ILYTIX8kDk/1FX9xIHvn9XHDH9PL4+6rilS6ML9Net9UxriiywNor6nCZ+G8X2M+0kDAwY7okAESACRKAoECBxUBQfEydJBIgAESACUQjkWuhxVDamAABAAElEQVTwNX2LzIiRKBTzpw+SCPCMlIaKt1KEAUgBlA64opEh3/uLv3CFFS2tAZkC2EAugEToV4bh137hF1xKA4ormtzQz3JNOqVkLsUSRf5cXX3M3M1533KxRVpPKVmA6AKQBci20fk7SZMGB3cflEN7DqWVqR1rGMyAgydEgAgQASJQBAiQOCiCD4lTJAJEgAgQgdwQuKMr0mB3hDJNri/TxPlHH3wwNye0yhsCFoUAh6hfsGxZKl0BdQwqNRJhqxYhfGvzZhl929tcTQTlEwSpDCANkFnyka2piINNmtKAdfhtJQ9QF0G7OabaMGCv8oL62r9/f+okp59fSVulyIMKTZuY9IiJbC5Q6BATaH01HV2AdASNknCSJg1AFkCChIFTlviPMzfOyNT0lOxat6vEkeDjEwEiQASKCwESB8X1eXG2RIAIEIGSRMAVOtSVJ2oXvFsL8UWJFVkEWQAp15x71zYRrfUoi4oAGhJo2QO5di3VqKB2T5Nsb9JNZ4VWj1cuXpTx9yYduQDi4G/Kvy0f1mtIZ8CGAotVusAHibBFj8d0oY8X/X/5N38jy0dH5be++lX541/5FdXkKqc05kAJhJqvyH/8jsinWrQWg5ISiI4ISsdIuo0iyAJEFVgbRY2M8CUsusC/XurHF/ovyBt9b8jklIadqJA8KPXfCD4/ESACxYQAiYNi+rQ4VyJABIhACSHQ99BD8ia6Iujr6bF07YJcCh0mdAEKAWHwyHYsSymFgMD0dJs0NCRFP0rROoozBK0eQSJcvZrqggiCoV4N39BWjiu0I0ad1kFQLkE0YCFDItzUIozfbmmRcv39QEpD0zvfKR/85CflW5/97Azf0Sen5Cvv0PaLT6qFhjr83OoO+U//9/Py5O++kCEPXKFDkAVY50YUOmR0QTTC/pXXe1+Xa0PX5NrgNUmUJ+SBiQf8yzwmAkSACBCBAkeAxEGBf0CcHhEgAkSg1BDo3rBBLunK0RU4vH5dxnUBOYYS+2nJVugQRfkcYaCkAaXwEFi+XESDQFxbxqmppBZJvNtu8caNpLzxRoo4WLtW5L3IWVASANEI19vbZfPevdKvREGZRh3oFfnff/iHklBnKKxYrltCOyC8/1//a5nW+gd/+1//a24Pj5yHUd0QOaDkwad+SjMghlqk+ULzzEKHCG3wogsaEg2CDW0UWa9AsckiP7j2A+m81SkDtwdkYGxAhsaGZOUyZYYsvSPLvbxEBIgAESAChYMAiYPC+Sw4EyJABIgAEVAE1mkl/hNKFIwODcnU+Pg9NQuygeTSErIZ8NqiIgA+p7ExRR6cPSuyfXtSkEXSoy0WTp7UYoiaYbJpE/R3p4lohNrDh52iSmsZQNDKcVRJgzKNNKjQvAbryoD9+37912VaKzC2f+5zzjbrjy69moqaz5g98bUOad3bITKuqrK0Ok0abK7aLLs37Jbm9c0kDDKIzX5wR1mCMq16mdD/lZeVKwejgBq2s99OCyJABIgAESgABMqOHj1KzrcAPghOgQgQASJQiggc0ZoFwQgC1CmAvIz+fQHZnm6ryO4IAWCK6PTMmaSc0gyBGze0XIB+xJpx4LosKE8kW7R4QTIpsmbN3UiEbI/211rvYnrFCvc7VKYMRMJIBI0++NbTT8uPst1s1z5mB97+W1/TBHwlKdZ/V0AWQBBd0LS2yW2eJQ9zROB833npHu6WczfOORJh57qd8tSmp3K8m2ZEgAgQASKw2Agw4mCxPwGOTwSIQEEicEUXI5u1D31Q2NIviEj88259pfymrhafunBBDqB0/iyyWt9UYiNZMAtQRXJ59+5Uy8UTJ0S6u1PkAYohbtyodSkeyZ00wOO+WysuIj2lT4sj3tHfqUndJnQDifCovFOJg+/PQOWwnoFMUN5iVmk489vyOx96v7NjOsKscM1qsK1xm6tt8NbgWzI+hXCO7JIEk6QRR21MO8oOFK8SASJABO4TAiQO7hPQHIYIEIHiQ+CIfmlFeK0v9sbb1/F4dgRcoUN0RdBCdq52gb5u/l+K7bN3ooPegPUa3UAaUJYOAhMTbbJjh7gNdQ406EQ0QEBbMOYWZeAjgXaP1vKxV2thOBJBUxhAIpQf/H+1Z+IJ2f9766X7//oF2as3fkI3LFnvEgdPp052ffeu21P/h0YafFWayv450xHuopKXo6aVTYKaB2Xp/0U5TWoKinzjG9o6Y0ySzz5L8iAKKOqJABEgAvcRARIH9xFsDkUEiAARKCUEbm3bphX0R+Smrgp7p6dlXAsdjg0Pu03ShMHLCoilJgSxYYRBEJGld47iiLr2z4v4JMJrWkhhZP96+cx/bHTFNcd/4bty9Jf+SH8HRRsw/p/eeFqt8ZS+2T41qKkJfyz/ThmFT8ofyPfXqUn3KVne8XEZCmnP6DngYZ4RSIJJOn5c5Ic/BJsk8pSmMzDqIM8o0x0RIAJEID4CJA7iY8Y7iAARIAJEIAcE6s+fl3q1O6WRBre1nd5kzEKHOQxBEyKQQQBE1bASU0gzclEtWmTT78jxvs9/TP7uQ4g1OJe5Z8ZBV5M8uu5TclPTJ57QrVLjEiqef17AawwoedB96BBJhBmAiYse6L7VLSOT2mNTpb6yXtbXr5e9GxHfEU+SmrrkCANtvynY0Jtz61athBkdlRRvBFoTASJABIjAfBAgcTAf9HgvESACRIAIzIrAT2mHhJe1RV45wo89KUMJfcg6vN6lEIH4CEyuWiUV6d+jr33n09LU/LsyfuWKjCpRNaWVF4O/c9lGaFhxW65Lg1zrHpRlaojqG9iGdavSLg4blUTQ2o1yO00i9EFfwtEIF/ovyOTUpExMT8jYJPpaKh6JKpm6g96VcxCNSnIkAQqgIncFRVLxN4OpSnMAk7cQASJABPKPAImD/GNKj0SACBCBkkHAFTpctkye0uiCXKRMaxyIFrIr1zx0tE5k+8RcUKNNFAJnrv6dnOw6Ka2nW7XKolp9q1X+za/9lTMPkgZTf/+abHvPsEbBXJTBQW3loDIwUCMrlDBoaNDfy7op6ZSflB+t+rE8eG1AapR8qNcNlkYi4BgkwqY0iTCmxAGWyX0lGI2wddVWwXbmxplMscPqRLXsWKMFLOYgbQ8/LKKbNtUQOX1a5OtfTxEHAcJxDq55CxEgAkSACOQBARIHeQCRLogAESACpYRA3//P3psA13Wdd54f8ICHjQRJcAFX6YkUKZGUAHrvKJIDOzHc8aJ04oxr4ibS6Ep3dbrslmeSaKpSXT2lSaZqaoYpVyWdTNVMjRxWgPakxkk7kROnie4ksGknkR1TArhJlEhC4oZ93zfO9z/vnceDi/OAC7wH4C3/o7q49517tvt7EPjO/31LLCbvq/vBnAoAxhS8r2/VQIewLoBogMMIBidOFBIyPmuGCEComtFo++2X/0/p7O2Uzu5ONQHQwaf8EyCwZuV770llS4v0SK3s1mb7RP0QqjWEgV71HNdrW6p65XJVj4ztrZNTjzWY2sHXXpPJri7ZoabzVSoiqMd90hrBigioO5RwaRhVIaGnwESEU3tOGVaZ+oEQmY0f+IDI3/2dCDIrwAKBhQRIgARIYMsJUDjY8reACyABEiCB7CcwAf9xBDpU8+EBdTuYvX/fiAbT6obwUF8X4cM9TItTFBPoUFPkHX/ssRQtWE0CfgImI4f+7sxr0DwE10RWhNYfq4UB3Orn9QjsK23mk2RwTf3dvaHNalU6QLHnniqRnhReMpUqjqFUvvSSOUM8KNG6BbU2GFIRokjP+G13XRqwnNKENQI+XFmXhj69Li9glwZ9/DWXNnUzMZkV4KZAV4U182MHEiABEtgIAhQONoIqxyQBEiCBPCNQpa4Ius9KBjqE/7gRC7TOnr+tdakyJKzFJWFAg6ExBWOe/QKt8XFsRo471dUm0OGsBjpckpFjPDFgQjQ4HD0sOyI75Gtf/Jp3pt0NDcZyYEoFgJVKw8kGbwpGKyQUqQAQPXfODPFQRYIRFRGwhIheuy4NcG1wXRogIsAaYaa+ngEWV3oD3HsUDFwavCYBEiCBLSdQdOHCBYar3fK3gQsgARLINgKIzH5JTfCLAh9eU22Ms239G7keCATBgm95k9/wBm+meH1ZU+bd1hSNVniwzZ7TwGgUDiyNwjz/8Ikn4tYFauWSKiPHn/3Jrxo4TfVNEtsbM8dqtGA50NXXJQPt7dKrKT96VQ1D34rHY17BYLXx7H2ICCjzEBICIoJxadB7sFCAyIC4CAgTWkguDY3vvity7ZqI/k01pbZW5MwZaTt8OP7a87Px618X0Vgo8vM/L236+8BCAiRAAiSwtQRocbC1/Dk7CZBAlhI4PDIih5FDXMtK36Rn6fLXtaw++I+rKTiePUzB5h7HWgWDMGOzTWESaLndIq2d6oagAfaRziBVoEPEzIBg0HCqYU2gYDnQ27Vf2rGF79lv+lYsVkjzzyJfwvoLLBFQShPWCLMqHkx6XBqMiKD3IBwgwKLqFnmf6rERbkwIiqp/W0zMAoDSGCmiricsJEACJEACuUOAwkHuvFdcKQmQwBYRyGcrgxH1/+5X14Ah/XBv/ccv6/XPIjVaigLrgj160CogBSBWhyYAV4T2a38gnfdXD3ToZuT4DIJrppPGcyJFcIPQK1+5IYQEKyL0/d7vSenFi6Le+rJdAyz64iIEUz0iUWlpQoxYeabsv9uGvyXPPCONjz8uojFRTNH3vU3jVrCQAAmQAAnkDgEKB7nzXnGlJEACJJARAjbQ4fsIODc7GxcM9NvAGf1Qvxg20GFGVsJBCo0AAh2ajBz6DTQycvzW/6Pf1CMrAiILegIdgk+uZ+TYowEWJ198URBfYQCWCHreqVkagqkeg3ER4NIAX9L+PMnS0LZ9uyonerCQAAmQAAnkJAEKBzn5tnHRJEACJLB+Agh0+Ib6GM/29MiUuiX4Ah2uf3T2JIGlBGygw/cSmRFmNa4FRANk5Pjyz7wmf/DHL8Y7IACAlrP1Z8256UyTfOfGDZO+U7RvLmfkgIuECbDYEA/SiAe8p3EWEBvBTfUIVwYICNalAXERbKrHERUSevNERNDHYiEBEiABEsgxAhQOcuwN43JJgARIIBMEnlfRwAY5DAYnxDe8aZmBZ2KBHCNvCFxTU/VZPWY0MwICHfqEqrr9dVJXWycQC9xiXBLcijy4thkaKpubzdPAAuGBIyL4XBo0OoDJ0hB0aRhEfZ64NBgY/EECJEACJJC1BCgcZO1bw4WRAAmQwPoJmECH+q3uYUQlT1EQu8GKB2gCwaBY29dUwAubhQTSI2ACHV5uFVnQIIef+4vkYEGh6vfOfqegA2waawRHREDGh1kVWBY7OmSXZh4Jpnq01ggIroiQjnBpUMQySGuE5O8YL0iABEiABDJPgMJB5plyRBIgARLYEgKjGugQyc6GNNPBjIoGsxr0MHWys/gSjXWBXkIwOL57t9RoerTdlZVbsn5OmvsEWu626E5WpPWqCgYImj+lhzrq//43Pidf+dIj8YAZOfzvdVBEKFUXB7gzAOVEIktDVK9tgEUrIqDOujSMqpCQF6keT52KB1PUbC8sJEACJEACW0+AwsHWvwdcAQmQAAmsm4ANdNilVgKLc3PGdxzZEWbGxkygw29r9oNUWSHgP/6UigUoxxGlPksKszVkyRuxyjKGdFOLQIezCHSov3O/9X9roMM57QS7+hSBDpmRYxWozm3r0hBM9bigQsKQJ9UjYiMgxmSp3oc1Aj7gTSfiIkBQLM8xl4a2f/JPdNUsJEACJEAC2UKg6MKFCwjay0ICJEACJJCjBL6/f7/ZuE3Bf1yzJARNwVMJB9nwuK9pLnd3vdm81mzgtdVrsEKVycih752xbHEyciQDHSYWejgat3lpqm+ShlMNW718M397+7S0t3fpdTwaYyxWIc3NMPrPvQJrhDkVEfAkQZcGCAmwRECAxYgeEBFgjTBTXy9jOSYi6PJZSIAESIAEtpgALQ62+A3g9CRAAtlJ4O6OHXJYMw4Ey9uarvCpCD6GZ095XoPO2VgF7iYcK8Q3vNlcXizD9oYlVwhcVfcXBDpcKSNHdaRacEAsQMkWwSBXGK9lnbBGiJ47Z7pAREBmy3EICXoNNwb834XDujTUaH1xa6uJi4C/Ynnh0qDPwUICJEACJLDxBCgcbDxjzkACJJCjBF7Dt/caL8AtW7ER7zt0yLggHEK2g5AF68w2gSPk0tksCwkkAx0uaqDDzz6KVRAUqj772GdNdgSKBZv/JrouDXfOn1+S6hFWB8G4CBAO4NJQpWemetz894szkgAJkECuEaBwkGvvGNdLAiRQEAQQ6LBfRYtBPeA/PqPf9B5a4clh4g9rCBQKBiuA4q3QBFrutah9uwY6vNZqzibQYfxXbNkYNiPHS594adk9Vmw+gSPNzWZSm+qxQt2YSi9eFF+qRxsXIZjqETJl6TpdGmrV6mG/WjZMaH/8ytzXjA84z69zPPMw/EECJEACJLClBBjjYEvxc3ISIIFsJQBXhUt9fV6Lg43amE+qWDA+NSXvlZfLgooAEAxwTI+Pm9gFP8c0idn665Lz6zIZOVScGlJLFcQt+K3/69kVAx0iQwLEgiINjIjDZOTQ389cyMiRTzEO1vKLBxFhSg+4NOB65507sk0FBdelAW4NblwEpHpEIKz+NaR63KuiwT4VDVAgVHTr8UCPHXogxkKVHjM6HgsJkAAJkEBuEaBwkFvvF1dLAiSwSQQ2UziY3rVLynUT9iMVDox1gW7ckBVhDpuyQIwCBg/cpF+AApgmGehQMyPMa6BD93dvUYWrYKDDs/VnDZVdBz+dTN+JCogGuVQKVThw3yMIBygD7e0ruzRoG7g5wDwVSVpXc2k4rK4PlSpMQDDAcUsPJNkIltILF4JVfE0CJEACJJDlBOiqkOVvEJdHAiSQ/wQgGqB85OZNc04V6BDf8Mq+faYNf5AACLS8p+4EsAHXr4WbjoX7FhdC1bC6ttxRCwO4wMxqcM1gRg4rWNXtrzMxC5rOPBr7nYGBrErfmenfhJuDN2VyflIWFhWs7n7LImVycs/JTE+zpePZVI+VAZcGWCPs1+CvNkuDjYsASwQIAFG9H3RpGES9WhGgHvfx6wjR4D094AbhKwtqlRCh1YEPDetIgARIIGsJUDjI2reGCyMBEiCBOAHrP/6ZEyeIpMAJdI51SseDjnjcAezQtumhAQtxdPZ0yrnn4hH2V8L0zTd+T1qvtMqvf6FdZtQNxhYrFtjX5z59TiAcBEuuWRgE15/q9Vv9b0nfZJ8MTw/L3MKcEQweqiIzNT8lXcNdcmzXMTmxOz//H4SQYEUEywciwkRHh4zpOZjq0WZpQHDFI9phHC4N2m5eryE63NVjSg8WEiABEiCB/CFAV4X8eS/5JCRAAhkkkElXhTF1QZjUWAV3tm832RGeu3dvxZXC4sBYF2ir4uHhnPIfX/HBeDMtAi1vtkhrR+vSMWBHjl0cdmsqHtTtrpOmk01St2/phr+lWy0T9CvjJYEOtf1X/vmjDAl24ELIyPHKK28loIk0Nx+WWKxS2rvaZWByQL8tL5LSklKpLK2U+cV5U1caKZXD2w/Lge0HJLYzZlEVzBmiAMq8WgoU6XVUr601An79InrM6IFfw1k9evRYrdBdYTVCvE8CJEAC2UWAFgfZ9X5wNSRAAnlCYE7NwYc0WNx7lZWyqELADIIdqmAwrbEL/lxf/5wGQExVIrdvS00i0NzxPLYyuFxTI88OwtB5aRlQ8/ndgTSYS1sU3qvO7s7logEwzCUO2JKrfXjn/U55+e7L8uv//I+k5vjzcr3tfzbWBearYNia4yth2JGj2LNe2jSjGxX408yXRT9eeeXpJaux1gZwTzhec1w+dPBDyfuv331dRmdHZXhm2DArROHATfUIMAsqHgwlRARoV9v1gHgA4QC/Ym7BPfyq9esBwcEWiBF2XFvHMwmQAAmQQPYSoHCQve8NV0YCJLCFBA6PjMglz/yrbaxsoMM3NWCcyYrQ22uyIphAh4nNcLH6EK9UCskl4TUNyhc0kX+uhP80BX8/WjrUYmClgq97sTtDCHw9/uRvfle+WBGTXY99SeR7jpVCQiw4HD1sRisE6wLzoKv8mJidMO4J0Uh0iWiAbrXbamVxfFHujt5dZZTCuY0Nf/Rc3C0GAsA04hvo4+PX65HzS5zHQT1BXHhCD8Q8GNXjweHDFA2Uw1aWRs2oIThUyBYI2fpvVlvUlXa2cnWcmwRIIBsJ8NNZNr4rXBMJkEBWEHhxHR+ikoEO331Xvj0/Lw/V0gAb46LAN+hva/1qIkRWQOAicocARAMU3b29d+fHJuAhXiJ14u9/43O4FAgGTfVN5rrhVIM5F/qPW0O3ZHZxVipKK2RbKYJGLC2wMBicGjTCwtQsPfeX0omLBQizAReFhMdMsgle9+oBgxjdmhpxYY+e9929KzMqNkBC7VlDqkdtzpImgUZNwynXrqkJiNqA6L9DoqmHRa28RN3iGqurRU6fFnnmGWmDoMBCAiRAAg4BCgcODF6SAAmQQCYJfF6/Of82PpA5hd/wOjB4GYoA3BRwrLX88PI35KPPqsWBltMVp+XFZ14UigXLKU7PT8vswqxEi6NSXort7fKCzAqIdzCl/yFQYiG6KyynEq/BXzgYvFg7KsTstGVALQvG9VvtKj1gdeDGRUAGBvRBgMUqPa+W6tGOyXOaBC5fFlFh2wgGO3caSwNRyy/RbCnSo9EpVPA2QsLTT6c5EbuTAAnkGwEKB/n2jvJ5SIAENpSADXR4V7+ZmdGAh6sFOsRiCs1/fEPfgAIcvKO7Y+lT79eXJjeentXt3uS/W9oi+QpBNp8pmpTPf/FryTpeLCVghYNitQwqLcb2dnlBsMTFh4uyuLBoRIblLQq3Bm4L9/Vb6p36TXaFCgRDKhagDB9BvoVHpVzv7dA2cGfYrtdWRIA1AlwYSlVICKZ6RKLaUh2fJTMEGqfUYubBA4Wqv+c/8zPSduhQcuBGtTiQ69dFfvSjeB2FgyQbXpAACcQJUDjgbwIJkAAJrEIAgQ6HE4EOF/TbmFk175xRU9swgQ4/jw9oLCSQBoGf/MX/RU7+d78l//vXzsjoQ/VJxi4LFvN6aXZhKcaGxcEvP/45ie2OpWjBahBYeLigX7A+NO5ExUUwrl9eIkURDR3xUBYXVTxQAYFlKQGIB90wc1+hTOv9aZjBa5nVwKiTXV1GbNimIgLsPCAg2APWCDbV44yODauGfro0KIU0C9wPNGCv7NmzRDTAqG1qfdAIIeFv/1aku1sa9d8uuiukyZvdSSDPCFA4yLM3lI9DAiSQeQJd+oF3XD9sTeu3t2Ma7HAtgQ4zvxqOmO8Ekhk5VKxaVKFqRr8JnJ2clH/xmT+W/9j6mUePj69uVylTc/e0xe5VWhX27YgazCMGycNFlQYCrkWWDFI04j8IC7BMYFlKoDIWkykVAsKU3Q0NggPCAcq99nZBgEVYI6RyaYD8ekhdGnTLS5cGQFtv2aYxPDR+geDsK2WQbrTAXQEHCwmQAAk4BCgcODB4SQIkQAJBAj984gkp1lgFRfpNY5VaHmzTb2qQLWFBv7kZeO893Wzw28cgM75eH4FB3Xy9rx/o59ScGELBTF+fzIyPLxGq3JGfP/S8vHDwBWm/3i6jC6NyZ/aOOds2CIRYHVn5W2DbtpDPcEMoKS6R2flZmVv0B4SDoADRoKi4SEqK+NEp+PtSob+7YYoVDdAWYoM5NzebM4SEByoiHP7gB2VYUz1CnomooOC6NBgPHY9Lw6C2jdKlwXBc6Uebil6Nzz4rbakaIWAiMizov3VtKUS0VF1ZTwIkkP8Eii5cuLA0clf+PzOfkARIgARCERjStImdar4ZUeEgomabOJAhAekU7fn+t74lhZQ+MRS4kI0uq7nybfW39aVj3B3IQhFyyJxu1tLVIq1XW+U3vvBdk8LTZuQIPtSf/cmvypHoUv9xtw1EBAgGsb0xaf54s3uL1x4CyKrQM9Ej/ZP9UhWtkk/GPrms1Y2BG/Lj+z+WiP6//8VTX1x2nxUaW083/dj8p7I8cEWDsLxgibDY0SGLeoaI4Lo04BrJAyHjIMgiXBoQmHGQLg1KYe2l8dIlkRs31A1K/aCOH5e2559f+yDsQQIkkNcEKJvn9dvLhyMBEkiHQJ+aak6pmTgEA1gdWPHAnPV1jX7QOq65r1kyS6CQRIOWnhYT6LD1WqvItHLU+AWInYESFFTuv/09efD2xRVFA/SzVgYNJxvwkmUVAkd3HZXh6WGZm59T/AggsbxMzE1ISaREykuwXWXxETDCgN6wLggQElAgJKxHNEBfxE6I4NBriAjI3jCh1ghFeg3RwFojGBEhUWddGka1H1M9KqRVSiPSDr/zjsj77z/KppCIRbFKV94mARIoMAK0OCiwN5yPSwIkEJ7AQMJU820VEAb1G3BYGpSoD6gRElRM+ASiUK9SJtT0vErNzVlWJvBtdf0ohECSPfv3S60GHmu5rdYFl1UswFek2KvCQr7oEaOvfOkvzAs3I0dXX5ec/975R41WuXrlC6+s0oK3LYFrfdfkje431Dy+WD586MNyvOa4vSWXey9L70SvjM5oVoCyHfLJJ5ZbJCQb82LTCEBImEuICIiBYEWEMr3GgTp8O4ZjOiEiqCG+lOt1vpZGpFm8dk1E3ZxMqa0VOXNG2hKZLoLP3XjrlggsDUZGTMBEOXZMRF1F2mZng035mgRIgATM31NiIAESIAES8BCw33w/pyIBCoSEtzS+Afxp9ybqzI0UP3584oQUqdiAYGrb1fwTftIH1Ie0KvGNcopuBVmdr6KBzcjRpcE1EegQ8TEu6/Wux74kcjEhHMCZOyEaIC4BLAUgGDylvztusa4HiGkAEWGlQmuDlegsv1dRUiE4ZuZnpH+i3/y/Go1EZWJ2QrrHuwUpG7dHt0tNZc3yzqzZEgKwRoieO2fmhojw4Ld/28RFQKpH16UBIgKyNMAS4TG9hogAa4SZ+noZyyMRoVH/Zsi0mi3p3xix/8YgCOIM7DSWl8bbt0UgNCgvk2kBogHiH1A0WA6LNSRAAoYALQ74i0ACJEAC6yAAEcEKC77uF/ftk4iagCI+ghsTAdcQEWCGjm+eaY3go5f7dSP6Ibxff0cG9X2GWIBgh3BBcDNy/P43Ppd8UAgGTfVN5nXDqYZkfaoLWB4ExQMIC7E9MQnTP9W4hVz/93f/3lgWTM5OmhSNCJgIsW92YVaqy6vlyZon5dl9zxYyoqx/drhJTL72WjLVI7I0uCKCcWnQp4AUDL0OcREgz+WTS0MjRANr5aYZgdqqqvQJH5VGCAmwSlBWAgH81CmBpQGDIT5ixCsSIAE/AQoHfi6sJQESIIG0CLTBpcGJi1CsH9CMiOAICR96++0lc6wmRixpzBdZTQCBDh/70P9gRANkRlhcWFgWswBBDlEgGKxnsw/hAIcRDFQ0YEmPQNdwl7EugIUBYhog/WKkOCKlRaWyZ9se+fhjH09vAvbeVAIQERBnAdYIKVM96oogIkA8wPZ6RIWE3jwPrtj4j/8oohmBBKkXn3hC2hjPQN95FhIggTAEKByEocQ2JEACJLBGAt9OmH+auAiwPIBwkBAPcP5JtTZgyS8CLb0t+k2fSDLQoQYn/8p/H49VEHzSoqEh2dZ/bV2CQXAsvs48gXcG35GFxQWTgvHE7hOZn4AjbioBKyJU6N/l0osXVRISgUvDanERICIMadvSPHFpaNS/O/JXfxUPgvhP/6m0Mbjvpv4ecjISyHUCFA5y/R3k+kmABLKSwD/cuSODiGugObEXd+yQIqQXVH/TkoSI8CmkvFqhvKnfAkW1b7kGrart6WFchBVYbdWtJYEOrziBDhFXLBGzwAY5xBohFqAUa1BNZONgRg6Dgz9IYFMJQERApgdYIuB6p/6t3hZwadDv4k3WBmuNAJcG5C7vz3FrhMY33xT5L/9FTSzUxuLFF0VOnjTXbRqcloUESIAEViNA4WA1QrxPAiRAAuskMKB+7SjvDAwkRQS8/uxjCNGVuvzD44+buAiwVkBgRQRYrEbQKxUf9quIMKUfcvcgEBbLphKY37VLhioq5D31GV7QD9qIXTCj7/G585rvHDoQMiQ4b8vZ+rOCQIY//9P/mxELarTvbg2MSMFgU982TkYCKQlAOECpjMWMkDDvydLgxkVAhoZKPXLRpaERmRPU2kL+5m9UAekXOXhQc7dWm39X8G/LkgJB4Rd+QdrUxYqFBEiABCwBCgeWBM8kQAIksMEEICRYEeEzmnEhVflbtVBwXRuKERdBDwRUtELCmStX5J5uZCv0W2zGeU9FMjP1oxo4rE+HGtIP1xAKbLDDGRUOjCWJ3nMDHUIwaDoTD3T4nRs3jFBgRYPMrIijpEOgvX1aGhqwHYyX9vYBfb3bvuSZBIyIgFSP0AEjapkQdGmIol4Pm+oRLg3IthPNYpeGRogAHR0isDpQAdqUoGAQr41bIlA4sDR4JgESSBCgcMBfBRIgARLIMgJ/qZHcSzQOggmoGIiNAEHho0ijxbKhBCZVLKi8edPMgUCHj3/4fxQIBTMasdwX6HD46h+btlYw2NDFcfC0CEA4aG/v0jHi5iGxWIU0Nx9Ja0x2zl8CcGlY1A33op4hIkBygisDDlzDncF1adDtuQzmuEuDPgILCZAACSwjQOFgGRJWkAAJkMDWEkB2hbcTJqKD+o2QjYsAqwMIBw0JX3nfKufUCqFU709oPIVsT/V4uaZGnh3E93RLSzZkl0gGOryusQvUS0TU6yRVoMMTagnyFHyGWXKCAIWDnHibsnKREBFQrEsDLA+sNQJEBLy2IgJcGkbVAiGfUj3qI7GQAAkUMAFYWbGQAAmQAAlkEYHdKhY8pyIBCjbRMj8vb2uMA4gIn0vUp1ouRIMfP/VU0qVhuwZhrFTf1qj6tGajMfZrmlMcLhhusc/u1m3WNawLWhHocF5nhGCAI+D+i7VALEChYGAw8AcJFASBooQrQum5c+Z5u37t10yqx0pka9AaKyLAGgERbkpVaDj48stJlwaICBpdQMqz2KVBl8dCAiRAAl4CFA68WFhJAiRAAtlBACICynNqaYCy2rfx39+/XyJqUg/LBARVHNZjZO9eKa6tlWFkeVAh4oCKCFVqcl/ope/QIXk/EegQbgiIXdD6IxUNEOhwUQ+gT4gGh6OHDS5aFxgM/EECJKAEHvva10xmhu72dkGqxxINPghJEakeXZcGCAlRFREOqYiA0LjTKhzAGmGmvl7G1ikiHNWxtuuYE9off65uqigBi4e5dY6nXVlIgARIYEUCFA5WxMObJEACJJBdBKyQkGpVU5rqzwZWNK4NSP+oVgrI0DCkB85PFrBogECH+MZvUC0GTFaE3t54sEMVDUxk8XgijCWCQVN9k8HdkBBvUrFnPQmQQOERQEaGyuZm8+CTSHGoZeC115KpHqsCIoJxadANf40exa2tglSPcHQK69JQqf0eV9EAmiYMo+b19Tt6Fq3Da1hFVOkxo0ICCwmQAAlkkgCFg0zS5FgkQAIksMUEpnUjXKQp/yIQDHSjC9EgKSTo65+8f3+LV7j507uBDq+p24fJiqBCwfT4uCzMzpqMFTYd2Ve+9BfyZ3/yq2aREAwaTjVs/oI5IwmQQE4SgIiAUvnSS+aMdI/31RoBsRF23LkjEBFclwYrIkA4gEtDlZ5XSvVYoePAagEBGCEcQOe8pYdbMNe4HqUUDlwsvCYBEsgAAQoHGYDIIUiABEggWwhEurvNUhbLy2V+5079BFtprAxsgMXV1vnmM8/Idk05uLi4KAcHBvLCpeFP/+EVEfXMaPUEOoRVhluKNEYEBQOXCK9JgATWS2CJNYKKCA8SLg2l6tJQpSJDMNWjjYtwQMWBIzopXBqQ6nFIr0v1er8jGsCj6r0VFrag6SQjFA9WIMRbJEACayWw9BPTWnuzPQmQAAmQQFYR+MyJEzKgG/93dNMvGhRxUIWEhyoizO7YIZ/XY6Xyun6QLdY4CMO6mUbAwncee0yi2rdcx6nVvN+5FBdhRK0rXnvn69J6tVWdfvWpZ/RAoENPgViAUqxuHuAn+/Z5WrGKBEiABNZPYImIoC4NJfr3FtYBEADGsMnXa2OBoK9xtnERYIkAEWFchQP8KUMMhSI9VhIN9DYLCZAACWScAIWDjCPlgCRAAiSwtQR2q5UBDhSICCgQEr6jm3+zMTY1y39MqUBgXRuKdeONeAjzaso/reeRI0ekWjM7HLsVN4ztV4uEPYFsCMtH3LwaE+hQU1DC9QCBDv/Xr38gLhTg8eH462RLPFt/1iwMggHEApTju3fLcYgGLCRAAiSwwQSsS0MwS8OMigcTKiIUwdVA12CzNEBIWNQ6zbFjhIPZEOtb1PgJtDgIAYpNSIAEQhOgcBAaFRuSAAmQQO4RsAKCPa/0BJODgxIpK0uKBzY2ggmuqFYIx27fTnbPBtFgTAMdTqpIcGf7dplVgWRWhZEZnLXuK5/+C/n9b3wuvt6EaADBoOlMPGDYd27cMGKBQDDQIx/Ltf5rsrC4IM/uezYfH4/PRAJ5RwBCgk31CKFgWEUEWBjAGqEaZz3wOm4jpReJckDPGt5V1M7MCA6JamPRYMUJW8fzxhNoxBQPHuibou+KivBwGRR1HWzDNQsJ5DABCgc5/OZx6SRAAiSQSQIf08He0g86RZoCEhkYSmB1oIcVEFaaa2bXLinTb/AndBOfrkvDSpkj5nSe0oRrAQIdziwsyOy9ezKtmSIW5uaMlYS7TmtdYAUD3IP1xUqWF27/XL2+NXRL3h18V+YXYG4hFA9y9Y3kuguWADb80XPn4s+vAsK8WhAg7SLcFGB5YAusEbbrUaNHTA+bTfbdz37WZFjQKpZNJNDY0SFy+bKqOyrvIJ2ypkMWWOepdVujpkuWT35S2jQNMAsJ5CIBCge5+K5xzSRAAiSwAQSwYf/JxDciAw8fytvqmiB6DGr951RIWKlANLj09NMmNkJ5RYWUIS6Cxleo0swFbuns7pSO7g5pfas1HncAN6N64Os0fCrW43s1dXLuhcQHZq1yy/V73433T8Qu+PLnvm2EDrSBa4VbLvyLC+7L5HW+WhjYB7wxcEPuj92X+6P3JVIckUNzh+wtnkmABHKRQH29zKpwYOMbINaBLXBb6NUDMRFsjARc1/3lX0qlHqMqQIRN9ajdWNIg0Kj/bsr16yJdXfpvmv6jVqNyDg78W6r/Joq60snbb0vjhz4kbWjLQgI5RoDCQY69YVwuCZAACWwGAYgIzyUyDkBEwLGSJcAPDhyQiLoJIEsB4gxMJuIiFOl5uwZcrNQPTdH+fmnpaBGIB0sKPvn264GwDPoFTeeDTnn5b16WptNNUldbJyOaWvK1G69K6zUVG/CJGUEOcWj5g//384IUiracSMRdeCogItj7+Xz+0f0fSfd4t4xMj8jIzIiMzYzJzgrNrMHPp/n8tvPZCoDAon6LHQjVYp56WjenU3qMa5rHykCqR4gHNksDAiziAz+yNEBEgEtDmV6zZJgAshpBHNCYQPKpT0nbwYNmgkb8u3TpkgjSIeOA64IK7SwkkGsEKBzk2jvG9ZIACZDAJhNYSTCwS5lSM0y4NCRdG1RAwGsIB8MQEdRc84MqHCwTDewAOONTLowGIB7cVfHgzsvyH/6nN+W3Xz0TFwo8gQ6rIzBVEIFgUIhigXl458dDVQngahLR/4qLivUbSv3AqjxZSIAEcpfAkG5EJ06flp137pj/nSEWDGNzGijlKh5UaxtYJmzXa1ggQECwR1RjJRxSEeExrYOIAGuEGbVmGKOIoEQyUCC2QyzQDEZWNMCobRpM2LgpaBwh6VX7kIS7XQZm5BAksKkEKBxsKm5ORgIkQAL5SWBKAxMWaVYDxEWwMRGMkAABQS0Gnrt7d2XRwGJZsBfx84W/+j+8gQ4PRw9L/f56Y5HQkHCvWNqz8F595OBHzEPfHLwpvRO98lb/W0ZEgIDAQgIkkLsEdjc0yEB7u3SreLBSgQXCtLapiMVkVoWDSTWZh9hQFRARjEuDigg1ehSrC8SMCgfQbOnSsBLd1e+1qUDe+NxzIvpvoahYsKRALIC7ggq7okGIWUggFwlQOMjFd41rJgESIIEsIxBRE82H5eWCff+cRo8uUlNMCAZGRMC3MFrgppBOgViA0lTfJLG9MXOkM16+9j1Wc8zENnh/9H2ZXVCzWRYSIIGcJ2DFgzAPgnSPaI8C8eC+ig4PVSTYkRARgqkeYYkA4QAuDQjbN6JCQq+6NNASAQTXVtrghhAQDRqR9lcz+SBAohzWf8cSLgxrG5mtSWDrCVA42Pr3gCsgARIggZwngCwFAxrjABkL8K3KoAoJiyokzKuI8Hn9FgwuCsvcFOJeBvEw4G60L4fGDy9/Qz767JfkNz/ym6a24VSDc5eXqQjEdsYEMQ+KEv+lasd6EiCB/CIAwcCKBngyiAiVzc3mISEiPFARocJxaUCqxwq9i+/Acdi4CAdURIAzBFwaICIgBWQpXRqUQvjSiH8PEdtAAyLK+++LaCwgeeEFaUN2BRYSyEECFA5y8E3jkkmABEggGwns1m9acKC4IsJ3VET4zIk6OV1xWu7M3pHR7aNxv3sE7UOQwxSiAcZBedDRJp//8GfjL/iTBNIk0NBQLrp3YiGBnCIAMQAuCHBZmFIBwFeCokGwjSsi2HuwREAKxzFN+QgRwbgx6GuICNalAZYIEBHg0oDSR2sEw2HVHwiUqCmOzaHpg0VTBosK7CwkkKsEii5cuMB4y7n67nHdJEACJJADBCAifGf8T6Xrdpd8v//7SwMdFq/+AGfrz0rTmabVG7LFEgLfvPZNmZmfkaf3PC0fPvjhJff4ggRIILcJwHoAAoI9ryYahH1aCAnzKiIU6XmZS4MOgjp86wiJuNBcGhrffVfk2jVVTvr06bXU1oqcOSNtcD9IURohHMDi4OrVeFBETVcsKsC0fexjKXqwmgSylwAtDrL3veHKSIAESCDnCbR0tUjr5VYxwQ/wRQtc7jU2lCkJ0QDCwBs33ohbIyyoNUKgdPV3BWr4kgRIgAQKm4CxHkAsgwxjKNJNbem5c2bURRUPhlVEwJ/qoEuDbocFsRGCLg2aN0CieejSYFIqTquJHISAsTHDxwRBnJmRRg3Q2wZrAk9pq6qSRogEepYrV0QuX1b1ReUXCgceWqzKdgK0OMj2d4jrIwESIIEcI9DSrUEQ9bNV6zUVDOCKADtYRE10rAtsoMMjURjALi2jKh7ApQGpFnG/+ePNDIS4FFGoV7Q4CIWJjUiABEIQgCXCYkeHTLz2mmzzZGmAJQIOBFmESwP+5A/moUtDI0SD8XF9Oi0avwfCgC2NGhC4De4JKUrjO++I/Kf/FM+68Bu/IW0p2rGaBLKVAC0OsvWd4bpIgARIIIcIdE52SkdPh7ReUbEAX7zAukBdOoPWBc8fel4WRvCRMnWBYIB4CCjMnpCaE++QAAmQwGYRgCVCRI8SDe43qOIBXCRsloaoLsIGWLRxEVB3SGMjwKVhVPvlS6rHtu3bRXAESiMyKWjWikbEMti3T9p8aYLRD6kYIyqvIC3jQ3qLBzDyZZYToHCQ5W8Ql0cCJEAC2UwArgidvYmMCda6wC444ZIAVwQUxClov9Yu7SPt5nWYHw0nG8I0YxsSIAESIIFNIGBcJF56ycxkszTYVI+VaokAqwMrIugWOZmlAQEWselAlgaICJpvQMr0Om8K0g7r8wviIOxWB5KPfnT5oyHLglopyJ490kbRYDkf1mQ9AQoHWf8WcYEkQAIkkF0EWvrUFUGtNVuvt6YMdFi3v07qauuWBTVEOkVYEZz/3vnseiiuhgRIgARIYE0E3CwNEBG6NeMDUj2WXLxoPNO2B1waICQgLgIsER7Ta4gIsEaYqa+XsRwXEdrU4qAR1gQQB/r7pVHFATl61LgjNEIkuHkzfu/gQZHHH18TZzYmgWwhwBgH2fJOcB0kQAIkkOUETKBDuCLABQHWBThsoMPE2hG74Kuf+KpAOFitQDzo6uvyNoOlAUQGlvUTYIyD9bNjTxIggfUTgIiAMplwadipJvxVARHBuDRoG1goIPwN4iIgPkIuuzQYd4U//dN4FgVYIJTrU8I9AW4LCJ64a5fIU0+JnD7N+Ab6XrPkHgEKB7n3nnHFJEACJLBpBJYEOpzRaRG7QF05XcHABjpsqm9ac0wCuC4ga0JsT0wHlTX3N534w0vgu13fldmFWXl85+NyYvcJbxtWkgAJkMBGE4CQMKDWCIc/+MFVUz1CPKjSI1dTPTYic4JaVZiUjRALtm2LZ1RAEEVYGiiDNsQ3YCGBHCRA4SAH3zQumQRIgAQ2mkDLbU2j2KnWBRAJkHcLVgaBzzoQDCAWoNA6wGDgDxIgARIggRAEEBdhLkWqRwRWxKHf2SfjIvRqXIQhfV2aIy4NjZqmUZBhQTMtSGWltC2sHBRYH42FBLKeAIWDrH+LuEASIAES2BwCLXc10OH9FIEOE0twrQsoFmzO+8JZSIAESCCfCUBEQJlXISGi19aNoUzr7DVcGmCNAJcG5CLoz8NUj/pYLCSQ1QQoHGT128PFkQAJkMDGEggT6BAroHXBxr4PHJ0ESIAESCBOAEICRIQiPbtZGlwRAdYISPWYqy4NfK9JIBcJUDjIxXeNayYBEiCBNAmYQIdX1RXBDXQYGBNpFOv318vg4CBdEQJs+JIESIAESGDjCUBEGP3t3zaecsjS4KZ6hEUCXBpgiQAhAVka4NIwiPoMuTSU6fywcmAhARJQj9ULFy7A4oeFBEiABEggzwm09LSYeAWt11QwQEYEBDqEcIBPXYmSKo2ivc8zCZAACZAACWwmARtcEakeFzs6pFzP2zxZGmCd4Lo0IKrA4DpdGvapxUNta6sRLPBP5F0VD5A6clFTR85RSNjMt59zZREBCgdZ9GZwKSRAAiSwEQRMoMPLKhbgUxTEAg30HAx0COuCpjPxQIcbsQaOSQIkQAIkQALpEgimetyRSPUIywNrjeC6NGDTD5cGbPrDpnrco6LB3oRogJjA3YlDT6YU6Vg4IipKsJBAIRGgcFBI7zaflQRIoGAIINChaFDnVrgjII3ilB4B+zKIBSgUDAwG/iABEiABEsgxAtYaAS4NEBEq1RLBjYsAdwYcqIM7Aw64NEBEGNDrMr12S42KBrsd0aBXb+IIFggHJefOBav5mgTymgCFg7x+e/lwJEAChUSgc7JTOro74mIBrApsGsXipRRoXbCUB1+RAAmQAAnkPgErIsCloeTiRdkWiyWzNFgBwcZFsC4NEBHG9JiGC4IiOPDyy8YgD5YG/YkjFZnSCxdS3WI9CeQlAQoHefm28qFIgAQKiYAJdIi4BfjUg9gFOALFplF89ZdeDdzhSxIgARIgARLILwLWpQEiAgqyNBSrVQLcGKyI4Lo0QF+HJ9+iHhANYKh3T4+VSvHZs3RXWAkQ7+UdAVjssJAACZAACeQYgZZedUUYV1cEN9AhYhg41gXVkWrB0VQf98NsONWQY0/J5ZIACZAACZDA2glUqrWBW0oTbgWzKh5MqojgS/UI0QAeffhnVP95XbXAPYKFBAqJAIWDQnq3+awkQAI5TyAZ6BCfcPD1yKwe+HoEJSEawLoAaRTrauuYRjFOhj9JIEmgvX1a2tu79HX8f5hYrEKam48k7/OCBEggfwkgNoEVERZ14z8MSwR93Ihe79AzrnEgLJBbDugLeP8h1SM3Ty4ZXhcSAf7uF9K7zWclARLISQLJQIfWugCfaKx1QUI0sK4IsC6I7Y2ZIycflosmARIgARIggU0gABEhagMcqoAwlwiKCOEAmrwtcGnYrkeNHjE9rKhw4wgFR8WRFaXxyhX1QSmTtuPHs2I9+boICgf5+s7yuUiABHKagAl02KOBDq+0xmMXBAMd4pONFggGdEWIs+BPEiABEiABElgPAbguQCzAP60w6HPLvL7o0QOxEdy4CHV/+ZdSqceIChC9mqUBQRZZNpdA47y+Oz/8ocgPfiCyb58IhYMNfQMoHGwoXg5OAiRAAmsjgECHnT2d0tmtvpPBQIcJscCmUTwSPVIQrgjX+q7J9Py0LD5clJJIiVSWVMqJ3SfWBpatSYAESIAESCAFAcQWRonoAYM+W4YOHzaXoxpkEQZ+2/VcoWcbYBGaflRFB2RjgP0BsjRARIBLQ5RCglLY4PLWWyLKX65f14iWM9L48KG0FVn/zQ2euwCHp3BQgG86H5kESCCzBDpHNQ3iUIf5tNH6TqucPXZWmp5sCj1JMtDhdbUugFiA2AX4iiMhFGCguv11JmZB05nw46JfLpdLDy5J32SfTM5NysPFh1JWUqaBqx4aEeHtgbdlX9U+qa2qldjOWC4/JtdOAiRAAiSwxQRGPvtZmVhYkJ137hiBYKq6WoY9rgjDKhyUj4xIuZ634VrXbUUEXJfqJvZgQkSYUeEAIsQgrRE25N1tnNQPS7dvizx4oJ+dPOmkNmTWwh6UwkFhv/98ehIggXUS6BxTseCmuhJ06WbfOjzu1MHUxrEVAoBu/JueXnmTb9IoXta2+GSRItAhrAtMoEMVDgqt9Ez0yODUoJRHyqW8pFwqo5UyvzgvU/NTMjE7Id0Pu00IbAoHhfabweclARIggcwS2N3QIAPt7dJ9+vSKA0+roICjIhaTWRUOkPZxh4oNVXod1Z7WGsG4NKiIgLpDKiRU6nlUhYQeiggr8g17s3FObURu3VITEbUR2akfvkpLw3ZluzQIUDhIAx67kgAJFCYBuBG8fOHl5Q8P20R89aCWAohN0PmgU8594tySdi3dmkZRbRuTaRQhOsChEpZ1Ces664pQSNYFSyDpi7f635Lx2XGJFEXkaM1R+cD+DySbXO29aiwR7o3dk279j4UESIAESIAE0iUAMWBKhYAwBUKDTfkI8eCBig5IzwgRoVJFBGxjrYiAjwX4bsBaI2DzBZcGiAj9el1OlwalsMbyd38nMjQkUl8vsnevyN27axyAzddDgMLBeqixDwmQQEETaOnQzb+vwL0AB75i0NJ5v1M+3fJpOfeFc9JhAx3iPpwicU4IBfYMwaCQxQIlkiyIZ1BaXGriGbiiARqc3ndaLt2/JLcXbhtxoWu4i+4KSXK8IAESIAESWA8BCAFhhIPDzc1J0QDzoF+l1qFAROhWEaFCxYNqFRHgcYhUj7BAgIBgD8RFgCXCY1oHEQHWCDO6CWaARQWySmmEYIAjqh+2jh4V2bFDZNu2VXrxdiYIUDjIBEWOQQIkUFAETODClZ7Y5nF6qI1UHHj522qdAJHAujSgL15rsWkUX/2lV+MV/GkIRCNRE7+gNOI3P4yUREy7hxoIaeEhfD1YSIAESIAESGD9BGBFAKsDuCykEhCCokFwtqCIgNewRIAH/rimfCzWa/yrBtcFiAjWpaFG64s1HSTiIuBfN7o0KARPaXz/fREcBw+KaAyKNv0M0AjRoIRbWg+ujFeRcsaRckASIIF8JrCqaOA+fEIcMJ8Y3Hq9ZhrFAJDAyydrnlRtpUiO1RwL3Im/nJydNBYJCJh4bJe/jbcjK0mABEiABEggBYHgxh8CAqwIUFz3BFOxyg+MhVKUcEUoPRd3XVxQkWBIRQSkgHRdGqyIAOEAARar9MxUjyDoFMQ1mFCzzQ99SNr274/fgOVBcbHGPMK3NSwbSYDCwUbS5dgkQAJ5RyClm0KIJ4VYsCOyQ1585sWCSKMYAsmKTVKJBt9///vSP9lvsizsqdyz4hi8SQIkQAIkQALrIWBEBN38715P5xX6QEiwIsKiigfDsETQ9talwbo1wKvRl+pRjfSlNCFGrDBNzt1qfPddkWvXRPr64muvrRU5bfzBYAAAQABJREFUc0YEAgFEgZ4e9QXRaBEanDIpGuTcU+b2gikc5Pb7x9WTAAlsMoG62jrp6uuS0YVR/cdMJzf/sifOsEVMUZ4/9Lw01TdJbG8sRQtWr0YAsQwGJgdkeHrYNN1budekZFytH++TAAmQAAmQQDYSgIgQTVgiBF0arIDgujS4qR7x/Xp/nmRpaITFAFIqwppgbCz+VsEFYWZGzS5GRAYGRO7fj4sH2q6xv1/a9iS+OIDYoDEljLiA61UyY2Tj70GurInCQa68U1wnCZBAVhA4Ej0i1TXVMjqn/0ip8G3iFuiljVmQapGxPTGKBqnghKj/0f0fSfd4t0zPT5uAibVVtXJg+wEGRQzBjk1IgARIgASyn4Dr0nDn/PlklgakevS5NKDOpnrMdZeGtkVNL/XMM9L4+OMaDGI8/mbBsqCqShqL1O/z5s34cfu2yKwGktLXjZUaKQKWCCoiiAaiNOWb35RGtPnEJ0zfeCV/ZooAhYNMkeQ4JEACeU2g5W6LyZJgYhzAssAT6LA6ooICLBE8pbWjlRkTPFzCVP3j/X+UoakhIxog00JNRY3s27aPokEYeGxDAiRAAiSQcwSONDebNQdTPSJ00nYVEtxUj6lcGpAhOppjLg1t27frA+rhFBMA8cCBuOUBgiBaYcG2gbDgFsY6cGlk9JrCQUZxcjASIIF8ItDS16ImcyKtV1pF5vTJYGGANIpwRnSKDXTYfr3d1EI8uDMbV79xjfsvHHrB6cHLMARuDNyQ3ole45oQKYoYoQCiAYMhhqHHNiRAAiRAArlOIBisERkfZlU4WOzokHI9b9Mj6NJQqnETXJcG5B0azHGXhjbNoIAsCr7S+NZbIn/4h3Hrg1/8RWlTywWWjSFA4WBjuHJUEiCBHCbQ0tUirVdVLIBIAOsCHLYkRAOIAShwXUCxogGuYXlwuuI0LpOl4WRD8poX4Qj0TPQY0aCkqER2VuyUjxz8SLiObJVTBOCSuqCfbJ99NqeWzcWSAAmQwKYSCIoImHzwtddM1ocdaqoPlwbNL5C0RrBZGlBnXRpG1QIh71I9IqbBjh0icHdgfAP8WmxYoXCwYWg5MAmQQC4RaOlR6wK192u9poIBhAJrXRB59BR1++sEwRFhPXD+e+cf3VjlCgERGRRxFUiB21d6r5jMCaiO1cTk2X3cVQYQ5cVLZNZCIO15iHRaKB7EOfAnCZAACaxEwKZ6rHzpJdPMujQc/uAHvakeEWARH2usNQI2gNMJEUEjBEh5jrk06JJZtoAAhYMtgM4pSYAEsodAy221LrisYoF+42n+VYVLgnWXS4gGZ+vPLotP0Pzx5jWJB9nzxLmxEmROGJkekeKiYpmcm5RbQ7fMdWxnLDcegKtclcCNG/Eg2QiUHdH/1w4dWrULG5AACZAACXgIuNYIbpaGkUCqRwgI9kCqR1giPKZ1EBFgjTBTXy9jOSYitKnJWiPSNmpsg7ZgvAMPK1atn0DRhQsXNBwlCwmQAAkUDgEEOhTN8GPcEfRsAh1COHBiF0AsQGk602TOvh9IywgXBZxTFbgo0OIgFR1//bW+a/Le8Htya/iWTM1NybboNikrKVM9xyo6j/ohJePJvSfl6K6jjyp5ldUEfvQjke7ueJwrZNlC5q2dO0U+8AGRj31s85be1TUpsZhG5WYhARIggTwngFSPKPMQEvQaGRnw1w8ignFp0DPq8DFoRoUDfG+Sdy4N+kws6RGgcJAeP/YmARLIEQKdk53S0dPxKNAhwhDDPNoRC/AoNtBhw6kGvAxV4Lbgigc2nsFaxgg1UYE0emfwHekZ75EH4w9kYhZvVOpC4SA1m2y9A+HgwYO4YADRQN1yjXCgX3RtqnCQrXy4LhIgARLYaAIQEiAiFCVEBJulwRURIB5U6ZHrqR43mmUhjU/hoJDebT4rCRQgARPoEHEL4IIQDHSY4GEDHTbVN8l6N/sQDhjHoAB/wfjI6yaAtNy9vSIIiA3rUsS0+gjjX66bJzuSAAmQwHoIQESYUxFhuqvLpHqEeOBaIiC4InzbccCloVczNAzpdWkGXBq2YW4dC1YOLNlPgMJB9r9HXCEJkMAaCbT0qivCuBPocEoHCFgXIPMBDogFKOsVDExn/iABElgXAf2cKq+/LjI7K/L00xQO1gWRnUiABEggAwSQ6hEiAgIt7kxkabAigiskwKUB1gjY7MPfvX+dqR73qVhR29pqnBAx3j0dT/8pkAkdb45CgpLIvkLhIPveE66IBEhgnQRMoMMrTqBD/AsUcIuHdUH9/nqTHYFiwTpBsxsJZJDAN7+pH0A11giEgw9/OIMDcygSIAESIIE1E4BwgGKFBJvqEYKBz6UBlgiIl7AWl4Y9KhrsTYgG+JjWnTj0ZMp2/Tlx9qxEVERgyR4CeK9ZSIAESCBnCbTcU+sCdUFIplGEdcGCHohdkBANXFcEBirM2beaCycBEiABEiABEthgAslUj83NZiab6hHWCBAR8NFquwansSICrBEQjQhZGg5oloYjem1dGgZRH7AeqFHRYJeKBviohrHUY80cekoWDX8TDEGVvMeLrSNAi4OtY8+ZSYAE1kkgbKBDm0ax/Vo7XRHWyZrdSGAtBBD4EHELJpEwXMu2bSK1tSKaWjxlocVBSjS8QQIkQAJZRQAiAiwRKlQ4WOzokHI9b9Mj6NIA6wTXpQEiwSBcEPQMcQGCAY7+xKEnbym9cMFbz8qtIUCLg63hzllJgATWQQCBDjt7OqWzW9MK2TSKdpxEdgRfGkW6JFhIPJPAxhG4dUtDiWgskTn9ZAjXA5SoRtXSFNssJEACJEACeUAA1giVjiUCHqlIhQP8yZ9QSwJkaUAwRWuNAEEBlgioO6SCATRleJGatI96hnCwUlnQMemusBKhzbtXpFGMKRxsHm/ORAIksA4CyUCH11vjWRHwr04g0GHd/joTs6DpTNM6ZmAXEiCBTBA4elQEx7Vr8WCHGLNMbVhPnszE6ByDBEiABEggmwhYlwa7ptJz58zlggoFQwkRwY2LAJcG6MgQDay1genAH1lLAGKBWygcuDR4TQIkkDUETBpFBDqESGDTKNq/X/hXRwusC0ygQxUOWEiABLKDwKlT2bEOroIESIAESGDzCRRpTINoQkRAXIQRFRHwsQ2WB+q9Zq7xOmGYllzgAb1a1AOhqkYTtVPqGoE+LJtLAILBp4Y06ea0fgCHKSEEBFocbO6bwNlIgARWJtDSrYEONcLOkkCH+FcEgkFCNECgw4aTDULrgpVZ8i4JkMAjAu3t8KyNl66uBenqwh+WeInFEkqkvozFInoUm7O9zzMJkAAJkMD6CLgiQpEKCLMaFBF/cR/9BY6Pu19PyKQACwXcx/dFs9XV0lNfb6wU9CXLBhNwrQs+9eMfx80HIRo81KSbxfqu6EGLgw1+Ezg8CZDAygQ6pzqlo7tDWjvVugD/kiA0L6wMrHWBIxg01TcxyOHKOHmXBEjAIQCR4Pz54PdaTgO9dEUEe93QUCo4WEiABEiABDJDYFEtDiDh4mMdPua5BbEOYGkAdwYTF0HPlRo74ck/+AOp0mMtqR61K8saCLiCQbLbzZsi770ncvCgyI4dIpGIvnGMcZDkwwsSIIHNJdByVwMd3k8EOoS0jH8xbHHEAlRRMLBgeCYBEghLIJVoAIuC5mZ8NI0XWCO4Fgmojb+eUfGARrIJTDyRAAmQQFoEZlU4wMc73YIusSIYOnxYitWyYPTiRXMfqR5tlgacU6V6VEN6KQ2ketQqlhAEfGLBkrrx8XhapMZG+evHHkuOSIuDJApekAAJbDSBlj51RdDkvK0rBDrEGuCOALEAhRkRDAb+IIG8JbBvXzyYolqlZrR0dSF2N1TJRyIBJoBVwfnzqI8Xa2VgX8fPo9oO341ROFjKha9IgARIYH0ERn/zN2Xitddk1507MpX4gz985Eh8sMFBkdOnzfWwCgcoO7VdlSMiJK0RVIA4qBka0HMmIRz0aarHMYoIhluqH0uEgUQjX525hfRIpWp1dwCRJ+IFbSkcWBo8kwAJbBiBZYEOXcvhhHsxxIIdkR3y4jMvUizYsHeCA5NA9hH4qZ/ayDVZc6bKxCT46LnUPSFxI3HCHycc+OCKhGIsJEACJEACmSIwrYLBg4RAkGpMtEGZ+/KX5b4GR0SAxR0JEcHN0gBJGAEXUYdUj/grT5cGhRAoPnFgpbqfhlsCjlkV3wcGpMgRDygcBODyJQmQQGYItPSodYEb6NCmUdS/RbYgK0L79XZjXRDbGxMcLCRAAiSQWQKwHOhzhoyLB/EKXFuxwGnCSxIgARIggYwS2N3QIAPt7aHGrIjFBO13J1pPqoDwQPtaEQEuD3BpgLyLv+I4Urk0qC2DRAvMGmElYUBxJIuvnUwoSWRTGB4W0VgHn0SARLyORmlxkCTHCxIgAUOgc0SDFQ50SP2ueqnbVbdmKi23W6T1sgY6XNCuEAtsJByMlBANIBggK0L7tXZ59ZdexR0WEiABEtgkAq7Jk3u9dPpYzFopLK3nKxIgARIggfURCCMeQDQ40ty8ZIJKratM1EFEgAAxq8LBYkeH7KqpkYhaHti4CBARcF0acGlA/O2BPHdp8AkBa66D2whSMd69K/LGGyLd3XELhJkZKbpw4YLmWGAhARIoZAItdzRQ4S0NVDjeGc+Fg7+uepx9Ujf4J5pWRbNioMNEb4gFKIWaRvHm4E0ZmxuTuYU5TTVULKUlpVJeUi4nak4kCPFEAiSQaQJdXTMaz+CBDot4B+FLQ8NuDYxov+sK348tSYAESIAEUhPAph8FG/+pxLWpSPwwVgZqaRC2YDyICrBEQJnXlI9Feh3Va2uNYFwa9DVcGvD9FSThUbVA6MkTEWHNwoA+vy2+vp+EcPCtb4n09IgcOyayZ4+xNhAVaigcWHI8k0ABEmh5U60DOtQ6wC34y4pkugnxoG53nZz7+Dm3hbnunEykUbyi/ZFXBzZiOCdiFphG+sNaF9jXhXj+wZ0fSM94j0zPT2vE4CKpjFZqVpsi83pb6TZ5YtcTcnLPyUJEw2cmgU0hcP78neQ8XV1uCpd4dSxWIa6FAUWDJC5ekAAJkMCGEXCFBAgAEA4yUSAkzCVEBDcuQpkOjgN1JYljOiEiDOBejrg0+Db8unzz2RJntwTb/vTbb4tcu6YefAkXvtpakTNnpP3o0WS3hgcP9DP9vHzXyaiAmxQOkoh4QQKFRaCzu1NevvCy/6EhyeKvapEeKiDU7amTpmebpK62Tkygw6sJsQBxx3AESqFbF7g4YGlwufeyDE0NSbQkKpWllQKxYG5xTvon+6W0uFSO7DgiB7YfkGO7VNllIQES2BQCXV3wpZIlgsGmTMxJSIAESIAENo2AFRHwvVbQpQEiAqwT8JEX95GlAdkZpjU9ZDZmaQiKALrkUGKBbffJRf1QD+uMy5fjFgW4geCHH/6wyNNPS7t+qWXbmgvnB+amcOAA4SUJFBIBiAYQD1YsEBBgeaAOTXWP1UnnhLaHUOAJdFgdqTYZEQrVFSEVxzd73pRbA7cU4UM5VnNMzuw/Y5reGrol98fuy9is5qdUvjUVNfKxwx9LNQzrSYAESIAESIAESIAE0iDgujQU6wYaggFcFyAguC4NVkTAx+CtdmlIRyzQ5XuFhU8ihsH4OG6LaBaL9l27vO2Cc8NKg4UESIAE/AQQ4DBROt9X0QARUeJiZDLQIdIovnDoBYntiTGNooXlnIuLimV72XYpKy1Liga4fXTXURmdHZWp+SmZmJ0wZ6cbL0mABEiABEiABEiABDJIoCjhilB6Lu6C2/Vrv7ZiqkcIBwc11WOVnjc71WNw0w4M6dQVF0MOiZf23RrDB4cWjGk/2sfvpp6HwoElxDMJFBABWBqsam3g45H4ywKxAKWpvsmkUGQaRR+seF3dvjp5q/gtiUZgDLe0TM9Ny8y8RqnVP9olxfxzvJQOX5HAxhBob5+W9vYuicWqBPEOEN+gufnIxkzGUUmABEiABLKWwGNf+5ogzgJSPVZo8L/SixfNJhqpHmGBAEsEHAjjFVULhQMqIuBfC8RF6NXgivq9vZQmxAi9TLukIwz4+mJBrmBgF+hra+s+fv++yOuva7yz7XLxU5+yXcyZn1SX4OALEigMAh3dHUsftDrxEmKkWs6bVIpLW5hXcEf48ge/bK4bTjV4WrDKR+DpPU8vq77ae1UGJgeMpcHOsp2yrWzbsjasIAES2DgCviCJGzcbRyYBEiABEshGAktSPb74osn2MKAiAQSFnXfuSFVARDAuDXoflggQERAXAaVvnVka7IbdDOL88NWHrVurWOBMq0mINAsRxAONh/BCLCYXjx83t18YHjYBJZe05QsSIIH8JwDXgjduvCF3Ku/I6MJo/IFTBDp0aaAtBQOXyNqvESixe6zbBEYcmR4xgsHOnTvlTG089sHaR2QPEiABEiABEiABEiCBdAkYEUE3y5qP1wgHGO++WiMgNsKOhIiAuAhLUj3qPdQdUiEB8RLCujSEFQHCttOp12xdgD622HkuPvGEvIBsCu++K9LWJi/89V/Hm0xPUziwsHgmgUIh0NLbogFRRK5WXlWZVJ/aBjp85Pq0Igq4ONTtj6urKzbkTS+BhYcLMrMwI7MLs2rYsaCC7qLMLcx527KSBEiABEiABEiABEhg8wlARECpbG42Z+vSABFhfyRisjAgS4MVEVZyaRjUEaJqmWA352bAxA9fHW756n116VgX+MYzy4IVRanKIUjZiCCKmpoRha4KBgN/kED+EzBpFC+3xt0QIBaoJVIyGkpCNEDsgh2RHQbGndk75py0SNBXNraBucEf6yIAy4KySJnsLN8pvRO9xlXh7thd+e5735Wfevyn1jUmO5EACZAACZAACZAACWwcAdelwc4CEWGio0PG9BxM9QiXhlKtd10akKhsEC4Nmu7Rt2kPW+cTC7CmsP1Xa/d9tToQPTDPc5O6aZiawuAUDgCZhQTylUBLt1oXaESX1msqGMAVQf+/N/ELIBQUxZ/aigFHokuDg52uOG0aWOEA8Q0QBJHWBnFu6fw8ueekdA13SWmk1IgHPeM9qlyHNPlIZ2L2JQESIAESIAESIAESyAgBZGmI4NDRICLAkHeipUWK9Dqq19YawcZFgEsDhAS4NIyqeIAAixNnlruq+jb22iUjrggYxxbfPEFR4u8qdbU4tNDiwJLjmQTyhEDnZKd09HRI6xUVC2ABb10REkKB3Z+erT8rN2/eXPWpIRjY0nCywV7yHJIABILYztiy1qjD8cP7PxQIB5Nzk3Jz8KYcqzm2rC0rSIAESIAESIAESIAEspeATfUY/Z3fMYtcUEuEYRURREUENy4CXBrw0bxU7x/QA5vxaRUR+n/5l2VAv9Uv12u3BDfyuOfb8Keq97X11fnmcdeBawoHQSJ8TQI5SgCuCJ29iTSL1rrAPktCNIBYgNJ0psmc26Pt0n693VyH+cG0i2EoPWpza+iWjM2NyRvdb0hVtEpO1Jx4dDNxhTSNSMVYXKQWB1bcWdaKFSRAAiRAAiRAAiRAAtlIwLcRj6glAQ6URRUIRlREmLl1S7aNjS1J9QghIar39//6r8tBvZ5V4WBcj7kPfEDGA9YIvnlWq6t84w3zPeL8Bz+IpSwpYcQC2wHzUDiwNHgmgRwk0NKnSqamT2y9rtYFEAusdYFj9Q7XgrrauqRY4D4mMiRADDj/vfNutfea1gZeLKtWzszPyMDUgFTN+IWDqbkpE/OgMlopx3bR2mBVoGxAAiRAAiRAAiRAAltMYLUNu7s8CAgVOLSy92tfkykVEKrv3ZMqFRFgjQBHAAgIxqVBRYRqPSJ/9EeyU/tgs97f3LzEpSHs3Adfekkq33zTfC+FcW6oGIF55n/lV2RBr1crwXmKLly48HC1TrxPAiSQXQRMoEO4IiDIqU2jGPi2GrELvvqJr4aKSdDV12UsD3AOFggGTMEYpBL+9aX7l6RrpMt0OLLjiNSU1xh3BLglDE4PytjMmHFTQLDE5x97PvzAbEkCJLAuAu3t09Le3qV94wprLFYhzc1LY7ysa2B2IgESIAESyHsCwc00Hng9dZMqHgz8t/8mcGmAiFCZEBFsXARs8BEnAUdEjyo9xlRIGPyX/zKUiLBPRYMKtTbA9gDHu3pofoRkgSNy5Ac/SL62F75nsZYJEB9YSIAEcoCAN9AhwrPavwh6aQMdNtU3rWmzD6uD5r3NxvIgtidmaFAsMBjS/lFdXi0l4yUyMDEgozOjxi3hUvcl/UcgYtIxlpeUy57KPbKvcl/ac3EAEiABEiABEiABEiCBzBLwbabD1mElvrbbnnxScKBMasyxvv/6X6VcxYPiixeNrB10adBY5xJV64HdX/2q1Or1XEJEGNLrMrUecOfYraJBqYoGC3oP24TbeqC/W0b1Rfmrr0rFr/yKqXb723ZWMLCvKRxYEjyTQJYSaLndIq2dal0AkQD/18PKAH8FUBJnCAYQC1DS2fA3f7zZjMEfmSPwZM2T0jfRJwuLC8ayAGccEc0BXB4pN2kZIRowKGLmmHMkEiABEiABEiABEkiXgG8znU5dcCOO9WG8KhUQcKBMfOELRkgYVpFg4t13pfrBA+PSADcGWCHgMC4NCREBXzshHgLKsFojRP/wD6XEEQ3e1/pJc3f5j3lYJOj8bkm1RrShcOCS4jUJZAmBlrsa6PD+yoEO12tdkCWPWFDL+IkjP2Ge98bADZl/OC8lRSXG8uDorqMFxYEPSwIkQAIkQAIkQALZTCC4kcZa06lD/5U247hvC+ZJWiJ8+tNGOEBd31/9lRx+/nkZUVHgoQoGbpYGIyKoAIC6PXqGJcGsHnDGe6AHMrGnKhAObAmzRgoHlhbPJLDFBMIEOsQSM2VdsMWPW5DTn9i9PKtCQYLgQ5MACZAACZAACZBAlhDwCQNYmq8+bF2YjXiqOWz9tuPHcSlV/+7fmfP2RGYEbPjHVERAzIIivbZxERAPAcELIRrAjmAl0UBvmzL99a9L5b/6V/blis9M4SCJiRcksDUETKDDq61LAx3apeD/fC1Io/jGjTfkxWdeTMsVIT4af5IACZAACZAACZAACZBAYRMIKwKEbQea6QgGYeeJfuhDggMFIsLcpUsyqeeIHtuxBj3mcNMpCK6IdI/wfO7TA5YJKDNVVSarQ5i5KRzEmfEnCWwqgZaeFhOvoPWaCgZuGkWETU0UN41i+7V2afpiPIaBvc8zCZAACZAACZAACZAACZBAeAJhNsgYzdcuVf1miAWY2zdPicY3wAGrg7mvfEVmVTyAcACBwC3bEi8QI+FxPZACsldFgxE9u8/qXustU2wdhQNLhGcS2AQCJtDhZRULEOYUkUogB9qYJAnRANYFTWeWigTpBDzchMfiFCRAAiRAAiRAAiRAAiSQtQTs5tddYDp1vk08xk5nTF/fVPO4z2HnhWiAbQWEA8Q5cMugvoDrgg2wiG1I2cSEHPzd35USPebVDWL0X/9r6df6bQlrhuB6KBwoHBYS2EgCCHQoMyKtcEeAdQEOOCChJEQDiAUoQcHAVPIHCZAACZAACZAACZAACZDAmggEN77onE4d+vs28umM6evrmwNz+4rbf0wtCMpVDIBwgO8oUcb2Ie+CpnKEMJBI9VilbdwsDRATouruUP1v/63s1OsFFQ6mtT3OU3rY9RRduHDBbmEwJgsJkEAGCHROdkpHd0dcLIBVgU2jiP+TneKzLnBu85IESIAESGADCHR1Tcj58/fklVcYsHQD8HJIEiABEtgyAu5G2l2Erz5snd04r2c89ElnHndOe+0bD2u8+Tu/Y+IdlI0jbKLI+N69tsuSc1SFA5RtfX1SodfIyADXBQgIJkuDnmFdAGPoRRUOcB77N/9GKBwoCBYSyBQBE+gQcQsgFljrgsDgyIrQcLKB1gUBLnxJAiRAAiRAAiRAAiRAAush4NtMp1OHNaQjGISd2zeH7/l94wXX2PPaa9Lz7W9L0WIwwoFvRJH9P/dzMvHWWzKr1gaVvb1SMTlpRASIB9YiAZkaICxASMDBQgIkkAYBb6BD2Ac51gXVkWrB0VQfj13AmAVpAGdXEiABEiABEiABEiCBgifg20yHrQM8X1vfRt7XLmzdWubxvaG+eVKtEUJA75//uW+YZF3Rw7izQe0/+2ey/+d/3tRjvPHr1+XBf/7PElXxYPR73zPe1FV6bQUEWCNQOEhi5AUJrI2ACXR4Ra0L5rUfoo0gCokNdJgQDWBdUL+/Xupq65hGcW142ZoESIAESIAESIAESIAElhHwbabTqfNtxDFpOmP6+qaaJ/iAYfv62lU99ZRMvP12cEixggFuWNHAXc+2kyfl+L//9+aZx65dk3E9Jn/8Y+nVc9XAgLFGoKvCMqysIIHUBJKBDm0aRQgGAesCiAUosC6I7Y2Zw1TwBwmQAAmQAAmQAAmQAAmQwJoJ+DbJ6dRhAe7G2S4onTF9fX1z2LmC57D9fe3cunF1P+hRy4MJtSIIFlgZHPiFX1hS7fa1N2wdBASUbadOMcaBhcMzCaQiYAId9migQ1gXrBDoEIIBXRFSUWQ9CZAACZAACZAACZAACYQnYDevwR6++rB1vo28ry/m9NWHrfPNE3yOVHP4+oadNzgmRASUnm99S2BV4IoGYce07eiqYFDyR74S6BzqlLpddet6PAQ67OzplM7uzuWBDhOuCMiK0H693QgGjFuwLszsRAIkQAIkQAIkQAIkQAJJAnajmqzQi3TqME46m/Gwc/vmcJ/BXvvG24g1Yj3VaimAYs++uUPXMR2jYckfeULAuBKo60Bnv2749RCN/3H22FlpOh4PSrjaY7b0tmjuEpHW62pdgKwIk3oghoET6LBuf52JWdB0JtyYq82Zy/f//s7fSyQSkY8e/GguPwbXTgIkQAIkQAIkQAIksIUEwm5escSwbX0b+bB9fe1Sze2bx4fSN6avr69d2DrfeKnWvdYxaXHge1dZl3MEWt5skdYu3eyPJJZepecKPTQbSSviEejmv+lk6o2+SaNoAx1CLPAEOoR1gQl0qMJBoZcbgzfk3ug9uTNyR6pKqygcFPovBJ+fBEiABEiABEiABNZBIOzmNWw7LMG3eQ7bP2w73xy+x/eNl6qvr23YOt+YYfv62vk4UjjwvcOsyykCL194Oe5O4K56Ql/M6GHFg8ut0vmgU8598lyyVUu3WhdoOyMswLrADXSYyI4AsQCF1gUGQ/LHwMSA9Iz3SN9knyyUL8jNwZtyrOZY8j4vSIAESIAESIAESIAESMBHwLdRTacOc2R64+xbj28O3/OhLmx/X7uwdb71+PqmWo+vrW9M+4wUDiwJnnOWgIlB4Fs9XAzG9Ej8lnfe6xSIDHUn6h4FOrSuCIE0ihAMKBb4oIq81f+WDE0PyfjsuMwtIFokCwmQAAmQAAmQAAmQAAmsTMC3UU2nzrfJ9Y2HVfnqw9b55vE9qW88X19fO19dqnWnM6ZvHt94vuejcOCjwrqcIZBSNHCfAAJCoqA9AiYaawRbmRANkBWh4WQDBQPLxXOGi8LQ1JD541teUi4lEf4J8WBiFQmQAAmQAAmQAAmQgBLwbVTTqQNU30Y3nTF9fX1zpHpDw/b3tQtb51tP2L6+dqk4+p7Rzs1P/T46rMsZAi0d6m6w1gIXBqcwjaIDY5VLxDSYmpuS/dv2S1W0SkZmbFCJVTryNgmQAAmQAAmQAAmQQEEQ8G1UfXWA4av31dnNqwvQ1y5sXaq5ffO4c9pr3zy+vr52YeswV6bH9I1nnyl4DralcBAkxNc5Q8BYDyBVYrDs0IpV9rMQC3ZEdsiLz7woTKMYBOh/faX3ihENIsUR2VWxS8rmyqQsUuZvzFoSIAESIAESIAESIIGCIhB2Qxy2HeAFN6+oC9s/bDvfHJgnWHzjperraxu2zjdm2L6+dngO35jB50vVzvalcOAjxrqcIIC0iKcrTsud2TsyunfUBDqUqC4dgQ7hfqCpGFOV//Az/0Fie2OpbrM+QODS/UsyMjsi1WXVUl1eLcd2HZMbAzekuMjJUxnow5ckQAIkQAIkQAIkQAL5TcC3UQ1bBzK+tnaj6pLztUunDmP75nHntNe+eXx9fe3C1oUdD2tKZ0z7TO7ZN7e3zu3EaxLINQK7ntglo9tVNECQQxz9eiCjwgqigd6Vi/cu4sQSkgCCIU7PTxvR4EztGdMrUhTx/uEKOSSbkQAJkAAJkAAJkAAJ5CgBbF6DG9i11rn9sVG1h0Vix3Pb4Z6tt+3WUmfn8G2MfeO5c/v62rW47dazHt/cvjp3nlRz23W6/X3Xvna+OjsPLQ58FFmX1QRa7rZI530Ncgg3BZtG0a7YCXQ4uqCCghZ7tk1whpsCSzgClx5cktnFWSkrKRMrGoTryVYkQAIkQAIkQAIkQAL5QsDdtNpnSqcOY2CjGizpjOnr65sjOKd9Hba/r13YOt96wvbFOn1tfWPaZ3LPvna+Ot8cFA5ckrzOWgItfRoEUVMrtl5pFUEGQJtGMfC3BrELjkSPLHkOKxzApaE6Um3uw82BZXUCl3svmywKSL04vzgveP3svmdNx97JXpmZ10iTat2B62M1x1YfkC1IgARIgARIgARIgARyhoBvA4nF++rD1oXdqPrGW8vcvnl84H3z+Pr62oWtw7yZHtM3nu/5Us3t67/S81A4SEWX9VlBoKWrRVqvqliAlIqwLsBhS0I0gFiAEhQMbDOIBSiIh4CC2AaMb2BQrPpjfGZcBqcHjXiwuLhozld6rph+E3MTMjw9bK6v9l6V4alhie2Kyck9J1cdlw1IgARIYKsItLdPS0NDeXL68+fvSHPzUsE5eZMXJEACJFCgBFbaQLpIwrZDn7VuVNczj28Odxx77Vs37vn6+9qGrcv0eKnWaJ/LPfvm9tWFfRYKBy5dXmcFgZYetS7QOAWt11QwgFBgrQsij5YHi4G62jppOtMk7dfapf16+6Obq1w1nGxYpQVvWwIV0QrZWbZT1OtMZhdmbbX3/HC1wBLeXqwkARIggc0n8Morb+mkcfU5FqvY/AVwRhIgARLIQgJhN5C+dngcX306G1XfeL46zO2bB/XB4uvv6+trF7bONx7WEba/r12qMYPPl6qdr943j6/O9qVwEKTN11tGoOW2WhdcVrFgQZcAsQAuCYmYBZIQDc7WnzVigbtIm04xrHhAawOX3srXJqZBrb/N63dflze63zCuCqf3npafOPIT/oasJQESIAESIAESIAESyFoCvs1iOnV2oxl84HTG9PVNNU+YeX19fXNgLF+9ry7smL6+vjrM7RsT9cHia+er883jq/P1LbENHz5cJQx9cHV8TQIZIIBAh6Ju8sYdQc8ypQeEA3wRlBANIBagwLogVYF4AEHg/PfOp2oitDRIiWZdN/ZW7TUBE2FosK9q37rGYCcSIAESIAESIAESIIHNJ2D3gO7M6dRhHN9mM50xfX19c7jP4F6H7e9rF7bOt56wfbFWX1vfmO5z2WtfO1+db461zG37Jy0ObAUGoYhg3w6eN4JA52SndPR0PAp0iPSJiGFgAx06sQte/aVXQy8BwsErX3jFiAddfV3xWAZ7Yqa/tUoIPRgbkgAJkAAJkAAJkAAJkEAeEXD3e+5j+erD1oXdqPrGwxp89b463zzuM9jrsH197Xx1qdboW4+vf9g633j2mYJnX1tfXTpz+/omLQ6wICsYuA1tXXDBfE0CayVgAh0ibgFcEFYJdNhU3yTr3ew3f7x5rUtj+3UQOLrrqLw98LbpyYwK6wDILiRAAiRAAiRAAiSwCQTcvZ2dLp06jJHORjXs3L457Prds2+8jVijbz2+ucPWpVqj+2z22je3ry7duVfqn7Q4wKLchlYwcOvQxtbjmoUEViPQ0quuCONOoEO4IrjWBfoSWQ9wQCxAWa9gYDrzx6YS+Nknf3ZT5+NkJEACJEACJEACJEACqxMI7uHQw1eXqt7XNp2Nqm+8VHP75kHbYPGN6evraxe2zjce1hG2v69dqjGDz5eqna/eN4+vLp2+XouDIAxXLHAX4NYHH5SvC5uACXR4xQl0iID8NtCh44pQv7/eZEegWFDYvy98ehIgARIgARIgARIggfQJuHs1O1o6db6NJsZNZ0xf31Tz2Gew57X09bUNW+dbT9i+vnZYv29M+1zu2dfOV+ebx1cXti/WsFJ/IxxYASDY0Fdv64IDu/Xug/O6cAi03FPrAnVBSKZR9AQ6PBw9bIDAugAxCZjhoHB+P/ikJEACJEACJEACJEACmScQ3MNhhnTq0D/sZjOdeXxzYG5f8c3j6+9rF7YunfGwZt88vjF9z+dr56tLNU/YuX3tfHW+uY2rgtvYFQB89W4dFm7bp6r3gWFd/hAIG+gQmRGORI+YB6d1Qf68/3wSEiABEiABEiABEiCBrSEQ3H9hFenU+TaLvvE2Yh4fQd/cYdfo65tq3emM6ZvHN57v+VDna+ur883jq8t0X3fdJRh8cXExWRdcgE8YsHXo5LYPU5+ciBc5TQCBDjt7OqWzu9OkUzRpFO0TJVwRwqRRtF14JgESIAESIAESIAESIAESWJmAu/eyLdOpwxjpbDbDzu2bw64/ePaN6evvaxe2LtPjpeIYfLZU7dJZT6ox02HhW7exOAguNJWQYIWB4CJ89bYOk7rt3XrfgliXvQSSgQ6va+wCZEWY1CMQ6LBuf52JWdB0Jh7oMHufhisjARIgARIgARIgARIggewn4O6l7Gp9dbjnq/fVBfd/a+nrGy9Vf988aBssvjF9fX3twtb5xsM6wvb3tUs1ZvD58NrX1lfnm8dXl+m+vjWjLhKJmFvJ4Ijuht5dxGoiAkZxH8SO49ahja/e1uE+S/YSMGkUEegQIgHEAk+gQ1gXmECHKhywkAAJkAAJkAAJkAAJkAAJpEcguJ/CaOnUob+7z8NrlHTG9PX1zRGfaenPtfT1tQ1b51tP2L6+dngK35hLny7+ytfOV+ebx1cXti9mD9vft24rFrj3jMWBb2C7qXcXt14RITi+HTv4MLbeXSCvt4ZAS7cGOpwIBDqERwsyIySyI9AVYWveG85KAiRAAiRAAiRAAiSQnwSC+yM8ZTp16O/u5/AaJZ0xfX0xpm8e1AeLr7+vr69d2Lqw42Ft6YwZfDa89s3tq0PbdOZOpy/mDhafWOC2SVocoNLduLsLsfXBB7ZCgtvWHcett2Pg/lrr0Ydl4wksCXQI6wIVDoyVgU2jmDgjM8Krv/Tqxi+IM5AACZAACZAACZAACZBAARBw90f2cdOpC+7bMKZvvFT1vra+Ot88dv3uOWxfXztfXap1+9bj6x+2DvP4xnSfzV772vnqws6dTt+1rHs1wcA+X8p0jGE2+u7DWBEBA7sw7DhuHdqspd62RT+WzBNouauBDu8nAh0idgHSKNriiAWoQhpFZkWwcHgmARIgARIgARIgARIggfURCO6PMEo6dejv7s/wGiWdMX19fXPEZ1r+M2x/X7uwdb71hO3ra4en8I25/On87Xx9ffP46lLN7Wvrq/PN7Vt3WLHAHc+4KriT2k26W4fJfPW2LviAq4kIaO+O747jq3fr0Ndtj9csayfQ0qeuCGPqirBCoEOMCusCiAUoFAwMBv4gARIgARIgARIgARIggXURCO5rMIivLlW9r627ubOL8rXz1aU7j53PPfvmSWeNvvEwX6bH9I3nPpd77Wvrq/Ot3VeX6b7uWt3r9QgGtv8SiwNUug/ibtB99W4d+tr27oOnEhHc9u44dgzcX2s9+rCsTGBJoENYF8w47RNpFCEW7IjskBefeZFigYOHlyRAAiRAAiRAAiRAAiSwHgLuvsb2T6cOY7h7rkyM6VuPbw47l3teS19f27B1vvWE7etrh2fwjek+m732tfPV+ebx1fn6Yi5fW19dqv52vfacjljgzrHM4gAT2M27u0Bbh/trqXcnQ99UQoId3x0b7X31tg733fZuPe6xxAm09Kh1wbjHuiCeWcM0gliAAuuC2N6YOUwFf5AACZAACZBAHhFoaCiX9vZHD9TQsPvRC16RAAmQQAYJuPsUO2zYOrT3tQ3urVK18/UNW4cxffOgPlh8Y/r6+tqFrQs7HtaWzpjBZ8Nr39y+OrRNZ+50+mLuYAkrFqCf73l8dUU/+MEPHqKDu6HHa1tSbcbXUp+q7UbOifWnmtc+W7afW97XDb8GKewc7JSmo01SV7O2VIctt1uk9bKmUVzQJ0UaxTk9bKDDxMMjM8KR6BHzqlBcEa72XpW5h3Pmdx7/k+I/lEhxRMpKyuREzQnzmj9IgARIgARIgARIgARIYK0Ewm4Cw7bzbeKwprD9w7ZLNU/w+X3j+fr62q1l3WHH9M3jq8PcvjGDz5eqna+vbx5fXTp9U63Ht+6wgoFvPb4691mSwkFwYt+mPtVGPBP1vvnsmnzj++rQfq31do5sObfc0M0+4g5gsw9Jp0oPpEHU4+zxs9L0VDzegNZ4y4qBDhM9CjmN4vff/770TvTK4sNFxftQivU/+z9ENBKV7WXbZW/lXjm195SXLytJgARIgARIgARIgARIIEjAfp5069OpwzirbeTsXOnM45vDjhs8++bx9fe1C1uXznhYr28e35jBZ8NrXztfnW8O9PfVh+0fti/mCZZ0xAKMFXaNxlUhOHlwALupDz6Q3aS79bYOY6ylPrhgO2eqcdyx0cbO69bbulRjoD5bSmd3p7x84eXly0Ecgm16qHDQekUFBbVAaDq9VDxIBjpM3E+mUUzELLCDQjBoOrO0r71XKOfB6UEZmRmRbdFtUl5SboQD++zRkqhAPCgpTvm/hW3KMwmQAAmQAAmQAAmQQIETcPcdLgpffdi64J4I44btu5a2vnncZ7DXvrl9fX3twtZhrkyP6RvPPlPw7Gvrqwv7PJnuG1yvfZ2OYBB2jZjLtl0WHNEuxD3bxqjbrA39WubEutw30goGbh3a+OptHe5vZWnpULcEX1GhANkPJLGXbe1oNakTz33qnJhAh1fjYoJAYMBhS0I0KGTrAovCPc/Mzxhx4MmaJ+UD+z/g3uI1CZAACZAACZAACZAACaxKILjHQId06tDf3fvgNUo6Y/r6+uaIz7T0p68vWvj6+9qGrcv0eKnWuPTp4q98c/vq0nkWzJRu/+DaMy0WpFqjj4XZjroPtNpG2h1kNREhuBA7tjsf2qxWn4k517oWtN/MAouDlAVuCzjgiq/uC533O+XT/9+n9f9efY3YBRAXnECHdfvrpK62ruCtC5TKsgIXBbgnbI9uX3aPFSRAAiRAAiRAAiRAAiTgIxDcv6BN2LpUbd09jp0z7Jhh22Fc3zx2PvfsG9PX19cubJ1vPKwhbH9fu1Rjus+G61TtfPW+eXx1me4bXDNehxUL0Na3Hl/dep6lBAOFEQCwkGAJLsKO4y7EigLom4n6MHNiLjtvJubEeBtZVhQN3IlNGMtExZRzIyEaIDPCC4dekOaPNzs3eWkJdA13SXGR/r7rf5PzUFxYSIAESIAESIAESIAESCA1AXcvYVulUxfcy2RiTN96Us1j57PnsH197TCGr95X51uPr13YOsztGxP1weJr56sLO3fYvliHb0xf/+Ca8TqsYOAbz1fnWwvmCdvWWBwEG68mAGACX3HH8Y2BPr4Nfap62xb33Qd1631zpmrvjpFqzlT17pxok8mS0k0hxCRMoxgCUqLJ7OKszC/My/T8tAxNDsmbPW+a18imUFlSKU/teSr8YGxJAiRAAiRAAiRAAiSQlwSCewY8ZDp16O/uWfAaJZ0xfX19c8RnWv4zbH9fu7B1vvWE7ZuKj2/M5U/n5+3r61vPWub29ffV+eb2rTusWIC+vjF9dWHXs1K7yspKMTEOMLG7MXYnXE0AQF9f8Y2Bdu6C3Dl99W4d+tr2br2tw/1MzIlxfOO7dWjjzovX6ZTqSPWj7vYSkg5iFqzwxfjzh56XFw6+IIWSRvERpPVdTc1PGdEAwREfjD+QsdkxY4Gw8HDB1N8YvCG122rlowc/ur4J2IsESIAESIAESIAESCAnCQQ/69uH8NWHrXP3JmsdD+3TmcfO5559423EGtMZM+wa3edyr31z++p88/jqMt3XXat7HVYw8K3HV+d7FswXtq1tB8EAB4qxOMBFcHC7Mbad0MaKCMH2ti3qfcUdA/ftOO6c7hhrqXfbYmw7jjunnQ/3U7V36+0YwfZh6tFnrQWb/665Lrn78K7InPaGYDC6+iixPTGKBqtjSrYoLSqV6rJqiRRFpCpaJRWlFeZ6ZmFGxmfGZWR+RBYWF+T1xdflY4c/luzHCxIgARL4/9l782fbrvI6dN5W96pBEqhB6PF0kYhlichCovHzs2OLJIUrKTtOJVSl0tgIYxfBTv6DV6/8U95PtjHGYExj2eSnVJJ6r4p6qbgpX14oG4G4kowRoIC4gIw6hPr2dm+Pve84d+x5x7f3nGetc84+53yT0l1rjzW+8Y059mn2nKy9TyaQCWQCmUAmsDMT0DUAZzgEg4auQ8bQdH5cD/bSo6vFdVfvuK3Y2HqRR50bz11vh7XOJerdWu9606sex94sgLbz6Pws4nGjgEd6XrvjAMCyhXHdlAtybawabFIfVcdpqJdIuwfXftBmT5w7HcXA4ZwUJxZpAF82PvP45C8pPD/5M4sP/cdSXpmw+UGH8mcUcTfCs6f8LgL+wsJu//OKyzLW6zdfeXPB3QX4jINbr7pVL5Vjjxwrj734WHn0uUfnvj7mSPkgE8gEMoFMIBPIBDKBTGDbJ6Cv6TkZh+Gawx1Wrzd6ap1eVO/6gFsPp+lqHa8Vc3rw0VrveJFmPb+I53DXx2Fj19ae+XjsDYPWuaB/xD1w4EDBf/VmAT3jOPfnGFWoZWGs4XIxrhpooDp4XA+nAY7qUEMxcHpwclHX0xN87UsdxcBZhoPDMf0zil+ZbBbgLyVgs+DVyX/4iwkYZzcN8NkFl+67tOjGwfde/d7aJgKuH7nyyLQk/2lP4JarbrHk26+5vXzh4S+U7z793fLsq8+Wb/3wW+WG195guQlmAplAJpAJZAKZQCaQCWy/BOrX75jBEEzXFJrGEE1XG/XRntFcolrXpxVzmq21jgfvTrOeX8Rzta6Pw1pr0bu13vnejM2CKJ9FvrFRwE0D51ux6VsVVGzZAthxa5PcRAAe8dUEz+snjjqRRg+uXPTjPLUn++F6xFecGjXf4Z955DPlzAtnyn98YLJhgLci4K8iYOMAGwVnNw34QYdvPPjGCXhu8DMQ3nL4LefAydkv3vqLc4/zwbAE8PaF/Xum3xKTpwZPTo5MIBPIBDKBTCATyAQyge2cgL525zyGYNDQ9cMYms6P68Fe9bG13vFaMefH1cKbwx3mNOu54bHjOQxc18dhrt7xHOZq0bserZsFqHOaDmv1s4jHuwp4rH1Hj5v/HKM258JYMTQgXk+SC3LlkxsZA646TgMc6kTaPbj2gzZ74tzpKAZO7eX+F+8v9z9yf/nM30zekoDPLnhh8h/Wo9XdBf/m1n8z/TOKd/1/d00utg3cbZB3HLRl1cM6uP/gdPOAGwg9tcnNBDKBTCATyAQygUwgE1iNBOrX6XA1BKvXCZFehLf2dn2gWQ+n52odz2HQd/gQzVa9em587Ho7zPVx2JBaeHL19KrH1g0Dp+cwN5fIj+NSExsFvZsFOq/p/71KMV7gglkbc1EMTi+u+k4bmqqPx/VwGuA4L4qBQ23FiUUawHt6qs4ff/uPy/2PTjYNJv+t3V0AAsbZTQN+NgGO9HLHTXeUo187OqXlP+Mn8JXHv1Iefu7hcmjfofKuI++aa/DQUw+VZ15+plyw/4Lpn2a8/vLr567ng0wgE8gEMoFMIBPIBDKB1U5AX+vT6RAMGroeGEPT+XE92Ks+ttY7Xivm/LTWOh7m4DTruUU8V+v6OCzSdFyHud7O99ibBejR6mcRjxsFPDrvrdjahyNy8YpCDWjZQr82Sh3FiUXawCM+rtXD+QMn0nC4Yqilxwhf1vOPHv2j6Qcd/vFX//jcn1E8ORHGWxHOjltff2v5sdf/WPml235pitQ93/WWd5UjVx0pd33uLpbYIzYY8m4DG81CEB+MiL+e8Nzkf/gwRHyuAcaDTz5YHnvhsfL8q8+Xi/ZfVF574WsX6uTFTCATyAQygUwgE8gEMoHVSKB+7Q5XDotwx9XX/Zyl47ViUW/Xh/306Pq4WsdrxdBvbE2np/PSc8d1WOt8xq5Vr3o+9oaBm5+bCzxEXH5mwRibBTrXPceOHTujAM+5qOVjHLmJoBjOHbcX79WuPfCx0xnDXzSfP/rWH5W77r+rFGwS4LML8B/fijA5xcBGATYNbr1m9kn+zo9i337i2+UvHviLcvyJ4xOpmdj0rQn55xdngQ7490+/9aflyZeeLPv27pvs6cx2dU6eOVleOvFSufDgheWNl7yxXHPJNeXIZUcGdMnSTCATyAQygUwgE8gEMoGNTMAtmoZg8OoWaEM0Xa3r4XLqqXXcVsz5aa11vChHN0fX22Guj8NcLfo6rsOi+tr7ZmwWoKfzs8g3Ngq4aVB7HuPx2h0HENPFq5oiXpvnIt1xodeD92pD3w3Vcf5Qw/movwgnF9fJv+tv75reXXDXVyZHftDhaRDO/jc5vOHgGyb/lvKZX5x8vsFkOB3FqQ3s+quuL2+68k3TzQNsGOBcudMH+c+6ErjqkqvK3j17ywsnXyivnHylnJn8D+PwgcPl8gsuL1dcdEVuGqwr2SzKBDKBTCATyAQygUxgYxPQ18vs1IqB77i6dujVdHoOg67rw356dPWu1vFasVY9+BqiqfPiuevtsKG9x/bdulkA324+DnMee3m8q4BH5rwRxz333XffZE17/k0HDoOBCOcivTbp+A4bS7vuz8c9/pyX+164r9z7/XvLXcfumtx6MWHggw5xl0F1dwE2DN57+3snF0r5h2/5h9Nj/U/v/Hv5db987BPA5xqcPI0nsZSD+w7mhoGPKdFMIBPIBHZUAkePvlzuuOPQ2pyOHn1y8vh1a4/zJBPIBFYvAbfAGoK1Ls6QxNh9XLqux0Z4bNV0fhyGuThNN0fHc5jr47AhtT2+WzcMnB+HublEfhyXmtgo2IzNAn0upx+OqKa4SFUMBQ4nVk9WF+mqQ75ikbbiDAjYMm1w3FANXKeOeqE/XCf+h9/+w3Lvw/eW+75/37m7C0DAOLtpwLsL7rz9zvLuH3v3WlYz0uxf9puW7ZnfbWBf9gSHWM1vwVGTY3kC+QGIyzNKRiaQCWQCOzGB3/iNr0+mNXu72pEjhyfnuXGwE5/nnNP2TkBfF3MmQzBo1OsBYEM0Xa3rgT5utNY7Xivm/LTWOh7m4TTd/BzPYVEfh7fWt9Y630M2C6J8Wv0s4nGjgEfnfSOxlfhzjBrQsoVx/cXCBXmkEYWnOrXGp7//6TL5/Lzyh3/9h7PNghcnKtUHHUL3moPXlPe97X3TFu++5d3TI/5xXrQfOOwZ8VUDHOaiOLFIA3iOTCATyAQygUwgE8gEMoFMYDskoK9z1a/DW7H6NTh0W2t7uK6PzoHnrrerdbxWDL3G1nR6nFN9dFyHtc5n7NraLx8P2TBo9eh46O+yAJefWbBVmwXMBsfz/hzjsgUtinRiLYtXDYj6qgFN6ihObFHPHm3ouEGNTz34qfLpY5NNg1MTFj/okAVn/zrC6w+8vlyy95LyntvfU/7Rrf9ozTdpPNJ7NB/2BJ+Z4DziO1wx1LqeiuM8RyaQCWQCmUAmkAlkAplAJrBKCdSvaeFtCIZ6fa2NxxhDNF2t6zHrNP+vqwXD1TtuKza2XuRxfnazR663w4bMBZ2G1tfeN2OzAD1bsyAPGwXcNKg9b9Xj6caBNqdZYlzU1k/SskWq8smFpuagswgAAEAASURBVOpTG7jjKwYOdRQnhus92uBjfOp7n5p+XsGn75tsGGCzgHcX7Jtenv7zvrfP7irApgEGNgw4Ii8OVwz19N7iW/mqQw1c78VRkyMTyAQygUwgE8gEMoFMIBPY7AT0dSt7t2LgO66+pu7VdHoOg67rw356dPWu1vFasVY9+BqiqfPiuevtsKG9x/bdulkA324+DnMeW3nsw7sKeAS+SuO8jYPanE542UIftRoaF7WKgUN8s7TRk97p5VPf+FT51D2TTQPcXYDNghOT//jRA2c3DX75Hb9c3v/O908uzAY1+JhHzofawInhvAfXTFCrPZ2OYuCzr+LEcD3CcS1HJpAJZAKZQCaQCWQCmUAmsJEJ6GtR9hmC1a+dx9B0fqI+7Mdja63jQcPhDnN+HK8VQ2+nCbwejuew1t5DauHN1dee8bh1w8DpOczND31aueRho2BVNwswH47pZxzwgS5SiemRkwOmXA2tZZHq+Kqt+o6L67049D/5zU+WY987Vo49fOzcWxH4ByXObhpgswBDNwymwOQf9bhs/uoP9cxFcWK4HuE9PSMd1QaHfRUnhus5MoFMIBPIBDKBTCATyAQygbES0Nec1ByCQUNfI4+h6fy4HuxVH1vrHa8Vc35aa+HXcZ1mPTc8djyHuR6od3hrfWst+tRjyGYBtFo99vK4UcBj7XsVH8/dcaAT1oWxM65cXCe/fmK5IFWcGOoiXPV7tKFJfWgfe/ZY+fLDXy6f/PInZ3cV8M8onv3MAvAxsFGgmwXUmF09/1/nD6xoPg5XDLXsGeFj9Iw8Rj3Bz5EJZAKZQCaQCWQCmUAmkAn0JFC/tmStw1sxfS3cqwf+kD7sp0enN8Sj00O/sTWdns5Lzx3XYc67w8auVa96PmTDoNUj+rVyydsudxdoljyfu+MAIBfpnJxiLHLHiK9fMMsWxspFD/J7tFEHnU98/RPlE/d8YvbXEIIPOvyVH/8V0AuPnDsw9UIfwN1Qf7hOHdUATh3FieF6Dz6k5xAvqM2RCWQCmUAmkAlkAplAJpAJuAT09SyvD8GgUb/uBTZE09W6HuhTD1cLjqt33FZsbL3IYz2/iDfEj6tFnyFZON9jbxZEHt183FzI410FPDrv2wHbj0nq4pUThHkugBVTPJqg8qkBrgaqPXtw1VYv0PiD438w/aDDT3x5smEQfNDhRXsvKhfuu7D86o//avmFt/3CdAr0skh7Spz8Qy4fu6PqLJu/zh1a1FecGK5HeE/PSEe1wWFfxYnheo5MIBPIBDKBTCATyAQygUxAXysyjVYMfMfV17a9mk7PYdB1fdhPj67e1TpeK+b04KG13vEiTZ0bziOew10fh41dW3vG49bNAnCdH4cNmQv78C8ibPfNAsyHY/pWBQ1HF4YapC6AI5yieoy42hN89lWcGK5HOPQ//tWPlz+4e7JpgA86fGny36somPyHcfaDDq/cf2X56et+utx4zY3ln779n671A2WRNq5jcP7KBa4e8bgeLfNXDdUnrhj0HU4M18foCR3tS33FwCGO8xyZQCaQCWQCmUAmkAlkArsngfp1IWY+BNPXsJriEE1XG/XRntFcXK3rEdU7bqumq3UYejtN4PVwPIe5Pg5rrYWP1vraMx63bhg4Pw5zXtCnlUvedn4rAua7aEzvOACBC8A6NOIMA1wuonEe4bj2sa98bCJcypce+VL59M9+eo6L66qjfdlTMfAV//iDHy/llVI+/qXJEXcXYMMAGwf47IKzmwbYLMD41Z/41XLTG24qP/qGH53TmF48+49qOzyap3qkhtbruWoA5/wjjR5cudCmF+3Jfrge8RWnRs1vwVGTIxPIBDKBTCATyAQygUxgZyWgrxU5syEYNPT16hiazo/rwV71sbXe8Vox56e1Fn4d12nWc8Njx3OY64F6h7fWt9aiTz1aNwtQ5/w4rNXPIh7vKuCx9r1THq99OKKGsWxhWIfOBenvf+33pxsFH/3rj05WxZOITk7+w3Hy3y//v79cPnjbB8s7rnnHWnaqQw1cjLx8+dkvl3u+d0/5+N2TzQL8+cT6gw7PfuDhFfuvKB/4iQ9M+/yzd/6z6ZH6kXYPrr4hvkx7asD8ozpOAyV8Llr8ffPJb5YbXnvDtJPjaz+Q2BPnjq8YOD1eyEVdjkwgE8gEMoFMIBPIBDKB7ZtA/ZqQM3F4K1a/LoVma20P1/Whfz263q7W8RwWeRyi6fo4PZ2Xnjuuw1wfh41dq171vHXDwPlxmJsL+rVyydvJdxdo/jyf+3BELiQ1TF0ARjjC+72/+b3y0a9MNgzw//pz04Dnk7sOsOB//3ffXz71858q77j23OYBjfAJwGP6wDl6/v5Xf39af8937zn3ZxRxEePsZsHr9r9u+vAD//sHynve+Z7puXp3+jofFJCvODFcj/Aebei44TTAcz0VA+e+R+8rL598uZw4daI89sJj5cKDF5bbXn/b0vn09Gz1wrxqj8ShkyMTyAQygUwgE8gEMoFMYPUTqF/PwfEQDPX6+hOPMYZoulrXY9Zp/l9XC4ard9xWbGy9yOP87GaPXG+Htc4l6t1a73o732NvFqCH8+j8LOLxrgIenfediq3dcYAJanBcvNfBcQGo+Ee+8pHye/f/3rlNA9wNgE2DyYYB7zjg+fv/6/vLB9/xwfLB/+2Dk4t+wMfHHpq8zeG5ydsdvjQ56gcdyp9RfPsb317e9sa3lX/7k/+2/Oe7/3N5z4/PbxioR/pGRzdP4I6vGDjUUZwYrvdog++GauC6ey7Q8wsPf6E8/OzD002DCw9cWE6dOVWefvnpcsG+C8pTLz1Vrr3k2nLjFTdOW9Cj+sYF4tqT/XA94itOjZrfgqMmRyaQCWQCmUAmkAlkApnAaiSgr/HoyGG45nCH6evMRZqu1mFRb9eH/fToNF2t47ViTi/yPVRT54bzqLfDh/QeUlt75uOxNwycR5cD+kfcnfhBh8y757j2GQcs4mJPA40WkuC+98/eW774yBdnGwXYMMDbE8yGgWIf+8uPlT1n9pRf+8lfW1sUsz8+F+FjfzXZLMDGw4uT//SDDs9uGmCjAJsGb/9f386ytU0DAPqkcz6KgUO8dZ6owVAdaigGDnHVBs4clU8urkdDdajxlw//ZXn0uUfLSydfKvv27iuHDxye9n3pxEvl5OmT5cmXnyyH9h1ak4x6Olz7QYA9ce74ioHDOSlOLNIAniMTyAQygUxg9yTwwAOT/19g8n8MnJ7cpbh/8n9jXHhhKT/yI7tn/jnTTGDVEtDXbfQ2BKtfT46h6fxEfdiPx55ax23FnJ/WWseDf6fJeenR8Rzm+jistRYeWuvVL883Y7MAvVrnQx7uKuCmAb3u5uN5f45Rn3Qu9hgeg8JC8ouPfbF85K8/Ur74/cmmAd6agAU+Ng64acAjrvGcxwn20c/NPgfh1/7er5WPPjg5n2wSfPQLk2PwQYcf/KnZHQo8TlSXLmjBcfOJcDfPiLseXPW5GFd/0GTmOHeDGsefPj690+DNl7+5vP6S15cbXze7s+CBJx4oT774ZHn8+cfL8yeft/PXntovwtkTfugb5xHf4Yqhln0VJ4brOTKBTCATyAR2ZgLHjpXyxBOTX/uT3/uT//+hXHDB7IhNhG98o5Srrirl6qtLOXJkZ84/Z5UJrFIC+jqMvoZg0NDXjWNoOj+uB3vVx9Z6x2vFnB9XC28Od5jTrOeGx47nMHBdH4e5esdzmKtF73q0bhagzmk6rNXPIh7fgsBj7Xs3P1745xg1VF3UfemJL5Vf+pNfOvfWBCz2+XkGbqNANgymmwxnE//oFz9aPvrVyWYBNhwmLyCmdyvwzyievbsAGwW6WXC2dHqov2C4qFXfINK74sRwPcJVv0e7pafTXuQF1ziOPXKsPPvKs5M/HrGnXH748nLTlTetLehvvvLmKe1z3/lc2b937p0odp46dxQylwjv9a061EYfhyumXnCeIxPIBDKBTGBnJPDYY6X88IelHJrcFIf/cKfBycndii+9NPnM48mHHj/66GyeuXGwM57vnMVqJlC/5oLLIZi+PuSMnd5G9GE/PbrerR5dbeR7iKbr4/R0XnruuA5zfRw2pBa+XL365XnrhoHTc5ibS+THcamJjYLcLOCz5I9rH47YujD+8H0fLr977+/ONgomfw5xuujHZoHbMFAM5/VAPf7jOLtpgLsQMHjEOf3hPBp84nFd+fpFwsWrYuAvw3u0oaf6Y2vj7Ql79u4pB/YeWPsmVX/o/zPX/QwO5w3nhRjIzneE1z2ZuWqglvotOLlRT+A5MoFMIBPIBLZnAl//einPP4+/v13K9deXcttt5+bx1a/O7kT42789t3lw7up4Z3fccagcPTqeXiplAtslgfp1GHwPwVBfvxYcqun8uB7o40ZrveO1Ys5Pa63jYR5O083P8Rzm+jgs6u24DnO9ne8hmwU9Hp2fRb65UcCj857YuQTW/i9pDZoLQNA07N+593fK7x6TTQO8PaHeHODdBTy6DYNz/dfODu09VA7vOVx+/ad/vfyrn/xXazi9qD9cJL5GrE6Ur1ydT8si1fFVW704Lq734C3a2DjA2z9OnD5RfvDiD8q9j947fdsC7kA4vO9wufWaW+d+AC2bv/qDX+aiOLFF81Hvy3pGOtqzxQs4OTKBTCATyAS2RwL4PIMDB2Z3GeimAdy/5S2l4G0M3/72bHPh+PGNe7vCb/zGj26PwNJlJjAwgfp1FeQcFuGOq6/3aM/xWrGot+vDfnp0fVyt47Vi6De2ptPTeem54zqsdT5j16pXPR+yYdDq0fHgwWUBLj+zIDcL9JlqO1/bOFC6PgFcAH743smdBl+ebBpgswB3CeBDEHXTgBsFPHZsGFy+7/Ly/r/3/vKvf/JfT23oItV5ASnCpwLVPxG3/oJiX8WJQTLCnb5yUUsdxYn1aH/l8a9MNwlOnjlZnn/1+fLADx4oh/cfLgf3HSynz5wu+HDE+5+4v1x36XVrdx04f4t6Oo+KtcxnjJ6Rx8gL+DkygUwgE8gEVjeBgwdnn1+AzQM3cCcCBj774BTeApkjE8gE1pVA/VoJIkMw1OtrOzzGGKLpal2PWaf5f3tqHbcVc35aax0Ps3Ca87ObPXI8h7k+DnO16OS4Dovqa++bsVmAns7PIt/YKOCmQe05H7clYDcOtBRPyu8c+53y4Xs+PPvgQnyegdswUEwFlpy/fPrl8hf/x19MWVxI1086cf0C4YYGCiPctVYurquO9mVPxcB3ODFcV/0xtLUntE+cOjHdOHj15Kvl9KnZ7gw+y+DQgUPl1OlT5YUTL5TnX3m+fP+575fPf+/z5afe+FOQWBvqDyA96jx1Pj24cqFNHe3Jfrge8RWnRs1vwVGTIxPIBDKBTGB1Enjzm/Gzv5QbbvCe8IGJ2FTAByZGHF+ZaCaQCejrJ6bRioHvuPoarlfT6TkMuq4P++nR1btax2vFWvXga4imzovnrrfDhvYe23frZgF8u/k4zHls5bEP7yrgEXiO9Sewv36iT1Vb/P/ys/+y3P3w3ZNPLZo0waaB3lHAcx7X6eMjn/tI+Xc/8+/mvvmWLQzrLxwuSCM8sqZ8aoCrX6zLvCgXteSrNnDqK59cXG/BT5w5UV459cr0zzDiTzD+3av/bvkHb/oHa9pffuTL5bHnHyvfe/Z75fvPfh+yC4d6dP5QTI8t/shFneNrP3DYM+KrBjjUV5xYpAE8RyaQCWQCmcBqJBBtCHz+86X84AezTYMrrlgNr+kiE9gOCehrIvodgtWv1aDp9CLccR3m+tC/HltrHa/Ho/PjNFsx9HaaOjeeO57DWnsPqe3xXa8jOZ/66Pw4zM0v8uO41MRGQW4W1M/C8Mfn3XHAL4AvfP8L5V/8P/9i9rYEbBq4P7U4cMOA9vHE84nmQlK/GFoWhqyHJjVwHuG4Vo+I2+sl4jt95cIP56o4MVwHjg9GhNbFBy8u1158LeC1eb7j2neULz78xfKtp75Vnnrlqbkf9KozLar+cf5AcV4UA4faLTi5qOvpCb7qU0cxcJbh4OTIBDKBTCATWI0E8FkGTz5ZytNPz/xceeXsTzKuhrt0kQmsZgL1ax+4HIKhXl+T4THGEE1X63rMOp3/b2u947Vizk9rbZSP0zx/dj5vV+v89PR29Q5zvZ1vrhXdNcUiPYe3+lnE40YBj+olz8dJ4LyNA8hONw3+78mmweTPIk3vNOAGAY8jv+/w7uN3l38/+R+GfjFxA6D+Ilm2MHQakfa0qflHNXB5vV7UO31DT/WpDdzxFbtg3wXlwgMXltccek256IKLyk1X3YSytcUyzi88eOHk3SSny8unXi74TIRbrroFsNWeXjD/qD9cpkf1ovPpwZULbepoT/bD9YivODVqfguOmhyZQCaQCWQCwxP40pdKefzxyV9Yxp9YnoyLL559psHtt88e1/+Cjz+/+PLkjkb8Wcarry7lmms27kMR6/75OBPYTgno6x717fBWTF97UbO1FvxWruvDfnp0eq7W8Vox9Btb0+npvPTccR3WOp+xa9Wrng/ZMGj1iH6tXPLy7gJ9ljb2/Lw/x/jb9/x2+dDdHzq3aYC30fPzC0bcMHjNvtdM/gbAnnLBngvKu29499wPHi72+AWBCKKFJLng6DcYcdVQHcVVGxw3Ir7riXqHKwaO86helE/uRQcvKhcfuLjs27Nvrb7ud6qcKnsn/8PYt3ff3Dcg9Z32tCD4x81fNVBGj4oTw/UeXPuhlr4jHdUGh30VJxZpAM+RCWQCmUAmsL4EHnpo8rnJkw9OPjG5Q/GVs39qGR+GWL0Dck38nntKeeqp2aYBPtfgta+d3Wlw5MgaJU8ygUxgkoC+lmEgQzBo1K+zgA3RdLWuB/rUw9WC4+odtxUbWy/yWM8v4g3x42rRZ0gWzvcqbxbkBx26Z2xjsbU7DvAF+Ftf/K3yof8x2TR4ZtKUdxfgOHDDAJsDB/dOXj1MxqX7Lp0e9Z+3HnmrPpz7oudir/4G4UJSv0HIhViEqw41FEMtcZy7oXzlRj17cNVGb+pT46YrbypPvPjE9HMOTr16avqnGG+/ZvZ/5XD+z738XDmw/0C5cP+F5Zarb1lbRENP9WttXMegzuzR+f86DbDoEefUUCzCycV15Sve0zPSUe0WL+DkyAQygUwgE2hL4PrrS8F/Dzww+QNM+AtMk4EPObxpdmPcDJj8++CDs7sS8NYE/BUFbBRg0yD67IO1wjzJBHZRAvVrFky9FYu4+lqKUbZqtvKg6/qwnx6dpqt1vFbM6cFDa73jRZo6N5xHPIe7Pg4bu7b2jMetmwXgOj8OGzIX9uFbEHgEnmNzE1i74+A37/7N8tt//tuT+wsnBgbeYXBwz8Fywd7JZsHZI6aEuwswzkx3JKan039uu+628tbr5jcOdMGoX2iK6xflsgVwjwZMOe1zjufPlIsr6/XS4/GyCy4rlx66tLx48sXyyAuPlGOPHCvYPPjqD75annjhifLUS0+Viy+4uFxxaPbpUj3amEPEx7V66Pw5d3AiDYcrhlo+zxE+Rs/IY9QT/ByZQCaQCWQCyxO4+ebFnMcem32ewf7J/3Vx2WWlvOMdi/ljXj1+/FQ5fhwvcs6NI0dmd+gdOXL2b0Geu5RnmcCmJ1C/DoGBIZi+ZtLJDNF0tVEf7RnNxdW6HlG947ZqulqHobfTBF4Px3OY6+OwIbXw5uprz3jcumHg9Bzm5oI+rVzy8q0ISG01xvSOg9/8wm+W3/qz35q9PQG/T+d/py51etHei6Y/1PTOAhRxswDnumGg+Pt+5n1rX0BceNZfaG4hSQza/MLCOTVwrjrkKwYOcdUATh3FieF6NCK+9mVPxaC3DKf2W9/w1vLYi5O/nPDM98qDTz5Yvvb418rdf3v3NGP8KUZ8BsKRy46U6y67bi2DVm340HmqR2qA4wb98Rp1VAPXqKM4MVzvwYf0HOIFtTkygUwgE8gE+hL4m7+Z/eUEVOFOg1tmH8HTJ9LJPnr0xGSz4PwNg0jmjjsOFPyXIxPYrAT0dQ97DsGgUb8+AjZE09W6HujjRmu947Vizk9rLTw7rtN083M8h7kePb1dvcNcb+d7yGYB9FyfVj+LeLyrgEfnPbHNT2DPT/xfP3Hmr772V7PNgsYNA2wUYFy8b/KpR2bs3TPbva8v6YYBr33+//w8T+eOXHTOgZMHusDUaxHudCJuhDsN9I5w9cXziBv1bMH/5Ft/Mv3rCc++/GzBhyYiX3y+wRWHryg3X3lzue31t7H93LFFWwuc90hD6/TcaeB6pONwhy3SGKNnpB95AT9HJpAJZAKZgE8Af3Lxm9/Ei81SbryxFPwFBZxv1OcaYNMA/9WDGwPRhgLuQgAn70Kok8vHYyXgFk3Qdngr1rqIc3o9vV0fl4vr42odz2GRxyGaro/Tc/MD5rgOc30cNnZt5HvIhkGrxyifRfPOuwuiZ2w18P1/9dWzmwaBnwN7DkzfcoDL3DCYrE+nI7qL4PSZczsQuolQ8/E2hWjoF6UuAPWLTRdvEe50lIv+1FGcGK47jUU4rtUj0tCeqGFfxYnhuuL4UEmMex+9t5w6fWq6cXBw38HylivfMsXrf6ijGsTAjXDnXbmoVR08rofTAEd1VMPhiqGWfMWJ4foYPaHj9BUDR/vicY5MIBPIBDKB+QTw2QfPPTe5uXHyF5temvyZZ/zlBXwGwp6zrymUjQ0FfDYCPjNhveP4cb9pAL1zb0/YO33rQr25gLczHD36ZLnzzqvW2z7rMgGbQP36AaQhGOr19Q4eYwzRdLWux6zT/L+uFgxX77it2Nh6kcf52c0eud4OGzIXdBpaX3sfe7Mg8tiaBXm8q4DH2nc+Xp0E9tdvS8BGgf5Hq/yU/uljfGAih9lEwCXeXVBvItx+5PaC/zB+5V2/Mj2eij5yeXr1/B823Eiov6G4eFOcGKT4BYpzauDc8RUDhzqRRoSjth7KxbX1eqFHfL4B/dW9hmpTj/rqfZk2a+ujauAadTgfYOyH8x5cuarT0lP5qrNeL9DLkQlkAplAJjBLAH854ZJLZn85AZsHGz2OH5/8iYcy+QTGMvk7j2X2Ac3sedddZ//sA4Hzjk9MEH/35HnUBDKBJQnoawpSHYZrDndY/bqmp9bpRfWuD7j1cJqu1vFaMacHH631jhdp1vOLeA53fRw2dm3tGY9bNwvAdX4cNmQu7MO/iJCbBUhk+4z9h/ccnrrlZkFk/fRkh+Ht1799enluE+Fswdve9La5zzGY05lsNGCzABw36i/qno0ELjqhq1/IXOwpBg5x/UZYphFpqwY41IlwcNxQPjXAU+/0HeHKBYd81QZOfeWTi+s9eK829N1QHecPNfTY4w91Ed/1jPiqAc56vaA2RyaQCWQCuzWBv/N3SsF/mz+wecDbGia3OBT8x4FzbCJwI0HPZ6+PyMxjJtCbQP36AfVDMH3tol6GaLraqI/2jObial2PqN5xWzVdrcPQ22kCr4fjOcz1cVhrLXy01tee8bheWzkOMOfHYc5LVO+41My3IkTPxPbA9/+nX/9Pa07fecM7p+dcGK1dOHvCRV2N9/AjDdXUL/YxNxHQQ7+Y6ZtfzPRAj44baQBXHWoswnGtHpFGr5eI7/SVCz/MRXFiuB7hPdrQccNpgOd6KgYOPbbg5KKupyf4qk8dxcBZhoOTIxPIBDKBTGCzEtCNgc3qmX12YwL16wFkMARDvb5OwWOMIZqu1vWYdTr/39Z6x2vFnJ/W2igfp3n+7Hzertb56ent6h3mejvfun5y1xVzmg5r9bOIx7sKeFQfeb69Etjz9NNPnzlx4sTa/xNd2+cCqMZ1YazXevmRjmryfNkmAnk8Ou3IX4Q7DehHfIdHGhFO//Ux4ruekceIu0ra9bz52HmM5jMG7vrRi9N3WPQ8LMLZI4+ZQCaQCWQC609g9hkHz0w+w+D5icir3UJ33vm/TD4LAW9zyJEJLE/ALZyGYK2LODgbu4+brevR6tHVRr6HaLo+Ts/ND5jjOsz1cdjYtZHv1g0D58dhbi5RPo5Lzby7IHrGti++Z7LYmW4cYAovTj6pCJsIGG7RFC2MHBcavfxIB1r1GGMTAZqRR4dH/hx3kbbTcVg9Z30c8Xu9OP5GamMOTt/50PnW504DHKfjsIi7CO/pGen0eoFOjkwgE8gEMoH1J3D06NOTDzp8vEsgNw264tq1ZLdoGoIhSC66NNQhmq7W9dB+PHe1uObqHbcVG1sv8sh56dH1dljrXKLerfWut/rl+dibBdB1Hp2fRTzeVcAj/eZxZyQw3Tiop4LNg2WbCKiJFkE9C6yIG+G1Vz7u2UiItKP5RLjTibgR7jQwpwjnfPUYcaOevbjT79WI+E4bc4v4Om+e92o4bYct8jFGz0g/8sL55jETyAQygUygP4Hjxyd/xmEyjh+f/EmH6fHFuTsKjhw5PPd4Ssp/MoEqAbdochjKHO6w1sWZq3VY1Nv1qaY3feg0Xa3jtWJOL/I9VLOeY9Tb4UN6D6mtPfPx2BsGzqPLIXpuwM0POuSzs/OPduNAp81NBGD5loZ4QduzkIwWhpFGhOvzpOcR3/V1GLQivEc70hlLW+fM8zH8Rb4jfCt6cr55zAQygUwgE8gEMoGNT8AtsIZgcOwWaEM0Xa3r4dLqqXXcVsz5aa11vChHN0fX22Guj8Naa+Gltd753ozNAvRtnQ95uKuAmwbOd2I7L4GlGwc6Zb6NYdndCGMtDKMFWYSrV5733ImAmkjbzclhvRrg9+hE/qDjRsSPevbgq6Tt5g6sx2PP3KEd8Xt6RjqRdoRDJ0cmkAlkAplAJpAJjJeAW+wNweCMiy51OUTT1boe2k/PW+sdrxVzflwtfDncYU5T58Vzx3PY0N7Oo8Oi3vTLY+tmAfhO02Gtfhbx+BYEHuk3j7sjga6NgzoS3o3ADYWeBVPERQ+3OIr4EV575eOejYRI2/mLfAN3OmNoRNrA3XA+wOv1EvGdfsTtxXu03dyBOQ3gPV56uGP1jDxGXsDPkQlkAplAJpAJZALrS8AtnIZgrYs4uB27j0vA9Wj16Goj30M0XR+n5+YHzHEd5vo4bEht5Md5b90wcH4c5uYS+XFcamKjIDcL3DO2u7BBGwcaFTcRgOG8d5E2Bj/SUJ96viqbCPDkFoEOAzeaZ4Sjph6LuK6vw6AZ4ZG+4ztsLO163nzc42+RF+fdYdDYyJ6LPOJajkwgE8gEMoFMIBOIE3CLpiEYOnHRpV2HaLpa10P76XlrveO1Ys5Pa63jwb/T1Hnx3PEc5vo4LOrtuA5zvelVj0M2C3o8Oj+LfHOjgEf1nOe7M4HRNg40Pmwc6H+9C6ax+JGOeuV5zyYCaiJtt2h0WK8G+E4n8hHh0HEj4rueqHe4w8AdQzvq2asNHTecTjSfsXDXE96cvsMi7iIc13JkAplAJrCbEzh69OXJX1k4Polg7zQGfCjinXe+cTdHsmvn7hZNDkNADndY6+LM1Tos6u36uCfSabpax2vF0HdsTafn5hf1dvWt8xm7NvI9ZMOg1aPjwY/LAlx+ZkFuFkTP2u7G92/E9PlFB21sIGC4z0Wov2i5ONIvcl1cKZ9caEf8CEdNPepv3mUbCZG286gY+tJ7jwbqVMdpgMO8VFtxnLuhfGqA53pGuHLBWeZR+eRG2hGuvsGh90gbHDdUx2mghh4j7V7c9UQfp6PYEC+ozZEJZAKZQCaQCezmBOrfqchiCIZ6/Z2OxxhDNF2t6zHrNP9vT63jtmLOT2ut42EWTnN+drNHjucw18dhrhadHNdhUX3tvV5v1Nf52Ok5zHmBRiuXPGwU6PqNPvKYCWgCG7JxoA3wRYhx6aWXTo/YSPjTr/9pefy5x8tLJyd/Eun0FJ7759SZU+XVk6+WSw9dWn7uR35u7tqyRR2/AVhEvuLEyHFH/cYecxMBvfSbnItR9QcOPTpupAFcdaixCMe1ekQavV4ivtNXLvwwF8WJ4XqE92hDxw2nAZ7rqRg49NiCk4u6np7gqz51FANnGQ5OjkwgE8gEMoFMYDckUP+OxJxbsYirv7uZYatmKw+6rg/76dFpulrHa8Va9eBriKbOi+eut8OG9h7bt64pOJfo6ObjMOexlYfe4PKuAh4jT4lnAkxgwzcO2IjHP//On5evPf+18j9/+D/LD1/4Ydl7Ynar4J7Tewr+h8XOK6deKc+88ky55pJrpl/Y//jN/5jlcz88dWGs30BcMKFIv4nIVwwc4jh3Q7/hezYRoEXtyF+Eq0enAW3OUzUUdxq4HuG4Vg/l4tp6vahH+oae6lMbuOMrBg51FCeG6z3a4LuhGrhOj1HPHly50KZ37cl+uB7xFadGzW/BUZMjE8gEMoFMIBPYSQno70jOawimv6MX6eHa2H3YT4+ux0Z4bNV0fhyGOThNnRvPHc9hro/DhtT2+Nb1A+fijs6Pw9xcIj+OS01sFORmgXsmEluWwKZvHFyw/4Jy0YGLymsueE3BnQUc2Dh45eQr07c0PP/y8+Wpl56a8k6fOT33g1cXQPwGgEa0wGrhRzr0pkf9IbBsEwF1Trv+ZqbHCHca0FY+NSJcNcBhXooTw/VoRPweL8pFH3rfLG305FzVC33gejScR9VAHXUUJ4brPbj2Qy19RzqqDQ77Kk4s0gCeIxPIBDKBTCAT2K4J6O88zmEIBo369zGwIZqu1vVAHzda6x2vFXN+Wmvh2XGdppuf4znM9ejp7eod5no737pOcNeJRXoOb/WziMeNAh7pI4+ZQE8Cm75x8K4j7yr4z43//q3/Xr7xg2+Ubz/97fLIM4+Uaw9dW37hR39hSuWCqf6m4CKo/kbr4ZOLRqqjuPNb/3BYtpEQaeucOB/0c7hqgEOPjhtpAFcdaigGDnGcu6F85fZ6cXzVRm/qOy6u9+Kq77Shqc8FHtfDaYDjvCgGDrUVJxZpAO/pGeloT3DYV3FiuJ4jE8gEMoFMIBNY9QT0d5h6dXgrpr9zqdlaC34r1/VhPz06PVfreK0Y+o2t6fR0XnruuA5rnc/YtepVz+s1gV7Tc+fHYW5+0Gnlkpd3F2j6eT40gU3fOFhk+Gdv+Nny3We+O70b4XWvf1150+VvKldcccX0Axbx4YoY+IwELvTwWL+xdLHDbxhwlvGVq/wIB8cN/aHRs4kALXqM5hPh6tFpQJu59GigzmkDd0O5uL5eLz0elYuebp4RTi6uq3f6Bq76yse1evRqOG3FoM+eET5GT/RR/WU9wc+RCWQCmUAmkAmsSgL6O4yehmDQ0N+vY2g6P64He+nR1eK6q3fcVmxsvcijzo3nrrfDhswFvYbW0y+P+rqfmDu6uTjM+YNeK5c8bBbkBx26ZyKxoQms1MbBnz30Z+X+R+8vJ06fKO++4d3ln9/0z6fzwxe/frgiwGV/pYELIHD5jYTzloVhxI9w6NZDf5gs20RArdOuf4BwTooTizSAO75i4FBHfQBnXooTw/VoRHzty56KQW8Z3qMNPdUfWxv69VB/uMa81Adw54UYrit/GT6k5xAvqM2RCWQCmUAmkAlsZgL6+5F9WzHwHbf+PRrxXG0rBk3XB3g9nKardbxWzOnBR2u940Wa9fwinsNdH4eNXVt7xmN9fe+uK+b8OGzIXNAPmnwLAo/qI88zgbESWKmNg4eeeqi8eOLFctmhy9Y2DeqJur/SsGwTARpceOk3LBd0uF5/0y7jRzrQqkf9Q2bZRkKkrR7pTzH0Ja4awDlX5ZOL6xGuOtRQTLVx7obyqQFe1LMHV21oUr9HI/LSqw0dN1SH/sBzHhUDh8+R4sQiDeA9PSMd7QkO+ypODNdzZAKZQCaQCWQCG5mA/v5hnyGY/q6kHo5DNF1t1Ed7Rn1dresR1Ttuq6ardRh6O03g9XA8h7k+DmuthY/W+tozHtev5R0HmPPjMOclqndcauZbEaJnIvGNSGBlNg4+9IUPle88851y7WuuLW9+7Zub51rfjdDzlgZ+07HZskVdxFdcNahbH/WHT88mAnSorz9EdPEW4c6jcqFNHcWJ4brTWITjWj0iDe2JGvZVnBiuR7jTV+5maaOPG84feOpx2TyVi1ryFSeG62P0hI7TVwwc7YvHOTKBTCATyAQygSEJ1L9noDUEQ73+XsRjjCGartb1mHU6/9/WesdrxZyf1tooH6d5/ux83q7W+enp7eod5no73/p63V0nFuk5vNXPIh7vKuCRPvKYCWx0AiuzcfDgkw+WZ195ttx0xU3lvbe+d13zrjcR8HkI/I8LbgjrN6MudPQbfL181UAv1cHjeugPpWWbCKhVfWrrfMDhnBQnFmkAd3zFwKGO84HrEY5r9VAurnE+ONe+7KkYOA4nhuuqP4a29uzRRp0bqoHr9Kjz1Pn04MqFNnVaeipfdaiB6704anJkAplAJpAJZAItCejvGOU7vBWrf/9Bt7W2h+v66Bx47nq7WsdzWORxiKbr4/Q4p/rouA5zfRw2dm3tl4/1tTkxd3R+HObmAr1WLnl5d4F7FhLbzARWYuPgv3ztv5QnX3qy7N+7v1x32XWjzB+bCHxbAzYPMMZ+S4MupPhNjT5cAOI8wnGtHvUPqmUbCZG2/oCiR8XQl3iPBupUx2mAw/mrtuI4d0P51ADP9Yxw5YKzzKPyyY20I1x9g0PvkTY4bqiO00ANPUbavbjriT5OR7EhXlCbIxPIBDKBTCATqBOof8/g+hAM9fp7Do8xhmi6Wtdj1mn+X1cLhqt33FZsbL3I4/zsZo9cb4e1ziXq3Vrvejvf9Wtwx4m8RD2cR8ddxONdBTxGvhLPBDYjgZXYOHj42YfLnsn/Lj90ecFfVhh7cANBP2DRbSKgr37zcpFWf5MvW9RFfMWpsWiu+kNszE2EIfNkJpEG8GieEY6aekRcfX5QQz+KE8P1CHf6yt0sbfRxw/kDTz1ynoqB04OTi7qenuBrX+ooBs4yHJwcmUAmkAlkArs3gfr3BpJwWIQ7rv4+Y7KO14pFvV0f9tOj6+NqHa8Vc3qR76GaOjecR70dPqT3kNraMx7r62x3XTE3F4c5j44H7YjL//MzNwv0GcjzVUhgyzcOPnbPx8p3nv5Oueqiq8oNr71hUzLBN2S9iYDGG/GnHqGrPzC4YaAYOMRx7ob+cOvZRIAWtfUHFBd0uB7h6tFpoJY6qqG408D1CMe1eigX1+gF59p3mRfHhYbqb6U2vLih/nCdHqP59ODKhTYz1J7sh+sRX3Fq1PwWHDU5MoFMIBPIBHZ+Avp7g7MdgunvLerhOETT1UZ9tGfUN6p1fVoxp9la63jw7jTr+UU8V+v6OKy1Fr1b651vfU3trhNzfhzmvECjlUtevhWByedxVRPY8o2Db/7wm+XR5x8tN195c7n+8us3Pad6EwEG3N0I+kNBF0D8ZkddtMBq4Uc60K2H/sBbtomAWqet8wGHHiPcaaBO+dSIcNUAh3kpTgzXoxHxe7woF33ofbO00ZNzVS/0gevRcB5VA3XUUZwYrvfg2g+19B3pqDY47Ks4sUgDeI5MIBPIBDKBnZWA/h7gzIZg0Kh/RwEboulqXQ/0caO13vFaMeentRaeHddpuvk5nsOiPq29Hc9hUe/au752rq/Vj52mw1r9LOLxrgIeay/5OBNYpQS2fOMAb1PAhyJeffHV5ed/5Oe3NJvWtzTUPwC4CKp/qHCB1cInFwGojuIunPoH4bKNhEhbPXI+6Odw1QCHHh030gCuOtRQDBziOHdD+crt9eL4qo3e1HdcXO/FVd9pQ1OfCzyuh9MAx3lRDBxqK04s0gDe0zPS0Z7gsK/ixHA9RyaQCWQCmcD2TUB/tnMWQzD9PbRID9fG7sN+enQ9Wj262sj3EE3Xx+npvPTccR3m+jhs7Fr1quf162S9pufOj8PcXKDTyiUv7y7Q9PN8uySwpRsH/+2b/6088/Iz5dSZU+V1h1+3cpnVdyP0/KlHTIY/HHDOhSHO9YcOF0fKVX6Eg+OG/oDs2USAFj06f7ge4erRaaCW8+zRQJ3TBu6GcnF9vV56PCoXPd08I5xcXFfv9A1c9ZWPa/Xo1XDaikGfPSN8jJ7oo/rLeoKfIxPIBDKBTGC1E9Cf63Q6BIOG/s4ZQ9P5cT3Yqz621jteK+b8tNY6HubgNOu5RTxX6/o4LNJ0XIe53s63vhZ214k5PYeB3+pnEY93FfBIH3nMBLZLAlu6cfCDF39QTp4+WQ4fOFwuO3TZSmdWbyLAbL6lYfaUcaGHR/oDd9kCuP7hSh3VgCZ1FCeG69GI+NqXPRWD3jK8Rxt6qj+2NvTrof5wjXmpD+DOCzFcV/4yfEjPIV5QmyMTyAQygUxg6xPQ3xl04zBcc7jD6t8tPbVOL6p3fcCth9N0tY7Xijk9+Gitd7xIs54fHjuuw1wfh41d6zwDG3vDoHUu6B1xsX7Af7lZgJRybPcEtnTjAJsFt1x9S7nowEXl77/p72+bLPEDAKP+gEX+2cdlizT9AUou9OofOlyoRfwIh1Y96h+mPXcjRB7pr8U3/FBH+dTA9Qh381RMtXHuhvLpA7yoZw+u2tCkfo9G5KVXGzpuqA79gec8KgYOnyPFiUUawHt6RjraExz2VZwYrufIBDKBTGBZAnfccagcPXqOdeedbzz3IM/WlYD+TKbAEAwa+jtkDE3nx/VgLz321DpuK+b8tNY6HubgNHVuPHc8h7k+DmutRf/WenrVY/36Vq/pufPjMOcFOq1c8rBRwE0D9ZHnmcB2TmDP5EX3me08gVXzjs2D6C0N6jWKXRd16+VHGqqn58s2EZSLc6cfzSfCnQa0Hd9hkY9FOK7VI/IBnuvrsIgLPNJ3Og4bSxs6bvT4Q32Px4i7kT0XeXTzTywTyAQygUxgfQm4BdYQDC646FJHQzRdbdRHe/Lc1Y/tsVUPnpwfhzlNzkmPjuewob2dR4dFvdUzzls3C8B1mg5r9bOIx7sKeET/HJnATkogNw428NnEJoL+17tgGosf6bipj7GJAF23aHQYuJG/iO/wSCPC0deNiO96ot7hDgN3DO2oZ682dNxwOtF8xsJdT3hz+g6LuItwXMuRCWQCmUAm0J+AWzgNwVoXcXA6dh83e9ej1aOrjXwP0Yz6OE03R8dzmOvjsCG18Ofqne/WDQOn5zA3l8iP41ITGwW5WeCescR2WgK5cbBJzyjfxuA+F6G24BZHPYsr6EX8CK898HHPRkKk7eYDfYePoQHtSCfCUVOPiOt8o7YXd/q9GhG/R7ueNx87jWiekY9evKfnWF443zxmAplAJpAJ+ATcomkIhi5cdGnHIZqu1vXQfnreWu94rZjz01rrePDvNHVePHc8h7k+Dot6O67DXG961eOQzYIej87PIt/cKOBRPed5JrBTE8iNgy16ZvmWBm4o9C6YxuBHGlEkm72JAB/O41iLUacdzX0R1/lxGLQjPNJ3fIeNpd07/14vju8w+OjJZNH8I/0IjzJIPBPIBDKB3ZSAWzQ5DJk43GGtizNXO7SPe+5cnyEenR76jq3p9Nz8ot6u3nl32Ni1ke8hGwatHh0PfqJ58zMLcrMgetYS3+kJ5MbBCjzD3ESAFZz3LpjG4kc6LqKeTQTUO+1o4RbhTgPaEd/hkUaEQ9+NiO96ot7hDgN3lbThxw3nMZrPGLjrR19O32Hg9+LskcdMIBPIBHZLAm7RNARDbm6BNkTT1boe7jnrqXXcVsz5aa11vChHN0fX22Guj8NcLfo6rsOi+tr7ZmwWoKfzs8g3Ngq4aVB7zseZwG5KIDcOVuzZ5h0Iy97SEC2AogVWLz/ScXGNsYkA3cijwyN/jrtI2+k4zM2bWMTv9eL4G6kN/07f+eBc3dFpgOd0HBZxF+E9PSOdXi/QyZEJZAKZwE5MwC2aWjHk4bitizNX24qht+sDvB5O09U6XivWqgdvQzTrueGx6+0wcIf0HlKL3vVo3SxAnZuPw5zHVh778K4CHmvf+TgT2I0J5MbBij/rvBuBGwq9C6YefsSN8Ci6no2ESLt3Ued0xtDAHJ12NPeI2+sl4jv9iNuLO23MM9JxGfRqOG2HLfIxRs9IP/Li5p5YJpAJZALbLQG3wBqCucUZMhmi6WqjPnX+rbWO1+Pb+XGarRh6O816fhHP1bb2HlIb+XG+WzcMnB+HuflFfhyXmtgoyM0C94wllglMfpZPXhjnn2PcJl8J3ESAXZz3LpjG4kc6LsaeTQTUO+3oSzTCnQa0Hd9hkY9FOK65MYaXXo8R3+EOwzx6fLt5r0cj8tKD9/ru0cacIn6UQeKZQCaQCaxaAm7RNATD/Ljo0rkO0XS1rof20/PWesdrxZyf1lp4dVynqfPiueM5zPXo6e3qHeZ606seh2wWQMf1afWziMeNAh7Vc55nApnAuQRy4+BcFtvqjHcg5FsaZk9btKAbayHpdBy26Iso4kfee/BV0o4y6PHYM3f0i/g9PSOdSDvCo/knnglkApnAViXgFk3w4vBWrHUR5/R6ers+LkfXx9U6XiuGvmNrOj03v6i3q2+dz9i1ke8hGwatHqN8XBbUzLsLomcs8UzAJ5AbBz6XbYfybgRuKPQsmCIuQnCLo4gf4VGYPXcjRNrOX+QbuNMZQyPSBu6G8wFer5eI7/Qjbi/eo+3mDsxpAO/x0sMdq2fkMfICfo5MIBPIBLYqAbdoGoJhHlx06ZyGaLpa10P78dzV4pqrd9xWbGy9yCPnpUfX22FD5oJ+Q+vVM87H3iyApvPYmgV52CzIDzpEmjkygf4EcuOgP7OVr+AmAozivHeRNgY/0ojC69lEgIbTjxZvEe40oO34Dot8LMJxrR6RD/BcX4dFXOCRvtNx2Fja0HGjxx/qezxG3I3sucijm39imUAmkAmMmYBbXLVi8OG4XHSpT8cbgkHb9dGePHd9XK3jtWJOD/1b6x0v0uS8eIx4Dnd9HDZ2Lb3qsXWzADXOj8OGzIV9uFGQb0XQZyvPM4H+BHLjoD+zbVWBjQP9r3fBNBY/0nFhjrGJAF23aHQYuJG/iO/wSCPC0deNiO96ot7hDgN3DO2oZ682dNxwOtF8xsJdT3hz+g6LuItwXMuRCWQCmcBYCbgF1hDMLeLgdYimq4361Lm01jpej2/nx2m2YujtNOv5RTxX29q7tRa9naard75bNwycnsOclygfx6VmvhXBPVuJZQLrTyA3Dtaf3barxAYCxrLPRQDHLY56FlfQiPgRjho3ejYSIm03H/Ry+Bga0I50Ihw19Yi4zjdqe3Gn36sR8Xu063nzsdOI5hn56MV7eo7lhfPNYyaQCWQCPQm4RdMQDL256FIfQzRdreuh/fS8td7xWjHnp7UWXh3Xaeq8eO54DnM9enq7eoe53vSqx9bNAtQ4TYe1+lnE410FPKrnPM8EMoFhCeTGwbD8tnU1NhLWu4mAifcusBzfYYtC3exNhN55RovUaJ4R7jJYxHV9HQbdCI/0Hd9hY2m7uQPr8bfIi/PusI3uucgjruXIBDKBTMAl4BZN4Dm8FWtdxDm9nt6uT+scXa3z47DI4xBN18fpufkBc1yHuT4OG7s28t26YeD8OMzNJcrHcamZdxdEz1jimcB4CeTGwXhZbmslbiJgEjjvXaSNxY90XLg9mwiod9rRgjHCnQa0I77DI40Ih74bEd/1RL3DHQbuKmnDjxvOYzSfMXDXj76cvsPA78XZI4+ZQCawuxNwi6YhGNLkokuTHaLpal0P7cdzV4trrt5xW7Gx9SKPnJceXW+Htc4l6t1a73qrX56PvVkAXefR+VnE410FPNJvHjOBTGBjEsiNg43JdVurtr6lIVoARQusXn6k48IdYxMBupFHh0f+HHeRttNxmJs3sYjf68XxN1Ib/p2+88G5uqPTAM/pOCziLsJ7ekY6vV6gkyMTyAR2TwJu0eQwJOJwh7Uuzlytw6Lero975pymq3W8VszpRb6HatZzjHo7fEjvIbW1Zz4ee8PAeXQ5RM8NuPlBh3x28pgJbH4CuXGw+Zlvu468G4EbCr0Lph5+xI3wKMyejYRIu3dR53TG0MAcnXY094jb6yXi9+hHGhHeo72R84/8RXiv70jH4Q6L5p54JpAJ7IwE3AJrCNazONuIPvWz4npshEen6Xq3YpiH06znF/FcbWvv1lr0dpqu3vnejM2CKJ9FvnFXATcNnO/EMoFMYOMTyI2Djc94R3XgJgImhfPeBdNY/EjHhd2ziYB6px0t3iLcaUDb8R0W+ViE45obY3jp9RjxHe4wzKPHt5v3ejQiLz14r+8ebcwp4kcZJJ4JZALzCRw9+nK5445D8+AKPHKLpiEYpuQWi0M0Xa3rEcXZWu94rZjz42rh0eEOc5pujo7nsKG9nUeHRb1r762bBahzmg5r9bOIx7cg8Fj7zseZQCawuQnkxsHm5r2juvEOhGUfsBgtdHoXWBE/wl3YY2wiQNfNyWHgRv4ifoQ7HYehZzQiftSzB18l7THm3zN39Iv4Y+QSaUd4NP/EM4FMoBRsHBw9enwSxd5pHEeOHC533vnGLYvGLZyGYK2LOEx47D4uRNej1aOrjXwP0XR9nJ6bHzDHdZjr47AhtZEf5711w8D5cZibS+THcamJjYLcLHDPWGKZwNYmsH9r22f37ZwAbhnDuPTSS6dH3o3ADQUumOpfDlzs8BcEisnFeS8/0oFWPepfkss2EiJt9cj5KIa+xFUDOOeqfHJxPcJVhxqKqTbO3VA+NcCLevbgqg1N6vdoRF56taHjhurQH3jOo2Lg8DlSnFikAbynZ6SjPcFhX8WJ4XqOTCATWO0E9HuXTodg0NCfNWNoOj+uB3vVx9Z6x2vFnJ/WWsfDHJxmPbeI52pdH4dFmo7rMNfb+a5fBzlO5CXq0epnEY8bBTxGvhLPBDKBrUsgNw62Lvsd1xkbCfUmAiZZv6VBf3FwsVP/Mlq2qIv4iqtGFLb+Au3ZRIAe9d18cD3CnUflopa5KE4M153GIhzX6hFpaE/UsK/ixHA9wp2+cjdLG33ccP7AU4/L5qlc1JKvODFcH6MndJy+YuBoXzzOkQlkAlubQP09CjcOi3DH1Z8pnJ3jtWJRb9eH/fTo+rhax2vF0G9sTaen89Jzx3VY63zGrlWveq6vdxSvz50fh7n5OR70Iy4/syA3C+pnIR9nAquZQG4crObzsu1d1ZsI2Dzgf1xwY5L6y0QXOvrLZ7181UAv1cHjeugv1WWbCKhVfWrrfMDhnBQnFmkAd3zFwKGO84HrEY5r9VAurnE+ONe+7KkYOA4nhuuqP4a29uzRRp0bqoHr9Kjz1Pn04MqFNnVaeipfdaiB6704anJkApnA5iWg36PsOgSDRv3zA9gQTVfreqBPPXpqHbcVc35aax0P83Ca9fwinqt1fRzmatHHcR0W1dfe9XVNfU0fOz2HOS/QaeWSh40CbhqojzzPBDKB1U4gNw5W+/nZEe70lwPfxuA+F6H+hcTFEX/RIAwu6HCufHKBR/wIR0096l+2yzYSIm3nUTH0pfceDdSpjtMAh3mptuI4d0P51ADP9Yxw5YKzzKPyyY20I1x9g0PvkTY4bqiO00ANPUbavbjriT5OR7EhXlCbIxPIBMZPoP4eRYdWLOLqzwg6btVs5UHX9WE/PTpNV+t4rVirHnwN0dR58dz1dtjQ3mP7rl+/cD7u6ObjMOexlYe+4PKuAh6dn8QygUxgtRPIjYPVfn52nDtsImDUb2nghgIXaeDoLyou0upfVOQrF7XL+KpDDdRFQ38Rj7mJgH7qfZlv5aKWfMWJ4Xo0zwhHTT0irvZEDfsqTgzXI9zpK3eztNHHDecPPPXIeSoGTg9OLup6eoKvfamjGDjLcHByZAKZwLAE6u87qA3B9GcBnTm9jejDfnp0vTfCY6um8+MwzMFp6tx47ngOc32HbxOCAAAwC0lEQVQcNqS2x7e+TuFc3NH5cZibS+THcamJjYLcLHDPRGKZwPZLIDcOtt9ztqMc129pwJ0IGNhI0AW9/lLiAgg8/mLC+Xr5qlHr4HE99JdzzyYCdOgxmk+Eq0dqQM/xFQOHeUUaEY7aeigX19brRT3SH/RUfyu14cUN9Yfr9BjNpwdXLrSZi/ZkP1yP+IpTo+a34KjJkQlkAssT0O85sodg0NDv+zE0nR/Xg73qY2u947Vizk9rreNhDk6znlvEc7VRH4e31rfWOt/6esRdJ+a8RPNu9bOIx40CHukjj5lAJrC9E8iNg+39/O0o9/UmAia37C0NugDSX4zRAquFH+m4sPWX9rJNBNQ77fqXLz1GuNOAtvKpEeGqAQ7zUpwYrkcj4vd4US760PtmaaMn56pe6APXo+E8qgbqqKM4MVzvwbUfauk70lFtcNhXcWKRBvAcmUAmMJ+Afg/pFYe3YvX3N3Rba3u4ro/Ogeeut6t1vFYMvcbWdHqcU310XIe1zmfs2tovH+trD2Lu6Pw4zM0Peq1c8vLuAvcsJJYJ7JwEcuNg5zyXO2omrW9pqH/ZcRHEX2IMhQusFj65qFUdxamrx/oX+bKNhEhbPXI+6ONw1QCHHh030gCuOtRQDBziOHdD+crt9eL4qo3e1FcucOalODFcj3DVX6YNHTecBniup2Lg0KPixCIN4D09Ix3tCQ77Kk4M13NkArs5Af2+YA5DMGjo9/EYms6P68FeenS1uO7qHbcVG1sv8qhz47nr7bAhc0GvofX0y2P9GoN4fXRzcZjzB61WLnnYLMBrNr5uq/3k40wgE9g5CeTGwc55Lnf0TOq7ETbyLQ38ZchAuZCMcPLqo/6S79lEgA576i92XbxFuHp0GtCmTo8G6pw2cDeUi+vr9dLr0fEVg5f1zl91qAE9N3T+nDt4kYbDFUMte0b4GD0jj1FP8HNkAjs9gfrrH/NtxSKufr8yv1bNVh50XR/206PTdLWO14q16sHXEE2dF89db1xz+JDeQ2rpVY/6OkJxd+7m4jDnsZWHvuDyLQg8Oj+JZQKZwM5LIDcOdt5zuuNnVG8iYML5lobZ087FJR7pC4Fli9f6hQR1VAOa1FGcGK5HI+JrX/ZUDHrL8B5t6Kn+2NrQr4f6wzXmpT6AOy/EcF35y/AhPYd4QW2OTGCnJKDfc5zTEKz+vhxD0/mJ+rAfj621jgcNhzvM+XG8Vgy9nSbwejiew1p7D6mFN1dfe8bj1g0Dp+cwNz/0aeWSl29FQGo5MoHdm0BuHOze535HzJy3xi37Kw31L00uvPjLEGFwQYfzXn6kA6161C8Ieu5GiDxyPi2+4Yc6yqcGrke4m6diqo1zN5RPH+BFPXtw1YYm9Xs0Ii+92tBxQ3XoDzznUTFw+BwpTizSAN7TM9LRnuCwr+LEcD1HJrAdE9CvZ/ofgkFDv//G0HR+XA/2qo+t9Y7Xijk/rbXw67hOs54bHjuew1wP1Du8tb61Fn3qUb82qK/zsfOCaw5v9bOIx7sKeKSPPGYCmcDuSyA3Dnbfc76jZ1zfjdDzlob6l+6yRV3EV1w1ouD1xULPJgL0qK+/9HXxFuHOo3KhTR3FieG601iE41o9Ig3tiRr2VZwYrke401fuZmmjjxvOH3jqcdk8lYta8hUnhutj9ISO01cMHO2LxzkygVVM4MyZPXNfz/RYfz0Db8X0+6xXb2gf9tOj8z3Eo9NDv7E1nZ7OS88d12HOu8PGrlWveq6vARSvz50fh7m5QKuVS17eXVA/A/k4E8gEcuMgvwZ2bAL1JgL+xCP/44Ibk9dfsrrQ4S9PcNbLV41aB4/roS8glm0ioFb16VHnAw7npDixSAO44ysGDnWcD1yPcFyrh3JxjfPBufZlT8XAcTgxXFf9MbS1Z4826txQDVynR52nzqcHVy60qdPSU/mqQw1c78VRkyMT2OoEsGFQD/1a5rVWDPz6ewpYa30rz/VAn3o4PXBcveO2YmPrRR7r+UW8IX5cLfoMycL51t/17jox58dh4DuPjruIx7sKeKSPPGYCmUAmgARy4yC/DnZFAthE4NsasHmAsexzEcDh4kh/+XJBh+v6C5hc4BE/wlFTj/qFxbKNhEjbeVQMfem9RwN1quM0wGFeqq04zt1QPjXAcz0jXLngLPOofHIj7QhX3+DQe6QNjhuq4zRQQ4+Rdi/ueqKP01FsiBfU5sgENiuB2dftnsn3jm4anH/HQf31TX8O1++bRTxX6zBoONz1YT89ttY6XisWeWmtd7xIU+eG84jncNfHYWPX1p7xuP6d7jjEnB+HDZkLekGTr49ys4Dp5zETyASiBHLjIEom8R2bADcQln0uAgLQX8pcpNW/vJct6iK+4tRYFLq+6BhzE2GsebqsoB3NM8JdBhFXe6KOz5HixHA9wp2+cjdLG33ccP7AU4+cp2Lg9ODkoq6nJ/jalzqKgbMMBydHJrARCdRfi+gxv3kw6+p4DtPvD/XruEOwqI/2xLnr4WodL6p33FZNV+sw9HaawOvheA5zfRzWWgsfrfW1ZzzW393uOjHnx2HOCzRaueTlWxGYfB4zgUygNYHcOGhNKnk7NgFsJNSbCJgs7kzQBb3+suYCCDz+Esb5evmqUevgcT30hUjPJgJ06DGaT4SrR2pAz/EVA4d5RRoRjtp6KBfX1utFPdIf9FR/K7XhxQ31h+v0GM2nB1cutJmL9mQ/XI/4ilOj5rfgqMmRCaw3Af06pMYMm7/jwH3GgauFhn4vzGvy0ezo6lsx12Ne/dwjp+nqHa8VG6IHp66P0zw3q3Nnjucw16Ont6t3mOt9zu25M/0dfQ71Z07TYa1+FvF4VwGP3lGimUAmkAn4BHLjwOeS6C5NoN5EQAzL3tKgCyD9ZR8tsFr4kY57WvQFyrJNBNQ77fqFBj1GuNOAtvKpEeGqAQ7zUpwYrkcj4vd4US760PtmaaMn56pe6APXo+E8qgbqqKM4MVzvwbUfauk70lFtcNhXcWKRBvAcmUBLAvp1pXyHnzmjjNm549Vf82A6XoQ7rsNcn5mr+X9bax3PYZFv58fVt2JOb35m5x45rsOG9B5Se87p/Jn+Pp6/Mv/IzcVhziOUWrnk5d0F8/nno0wgE1hfAnsmL9jMr871iWVVJrCTE8AdCG4ToZ5z9C2lCyytcfyIG+Gqp+ctGwnkR9rOH2oi3OlE3Ah3GugZ4bhWj4gb9ezFe/Q3UrueNx/3+EON8+iwiAt8jJ6RfuQF/ByZgFtgLcIeeuiF8slPPlz+w3+4MdwE4KJL012kuR6e66E6PHd9cc3VO24rNrZe5JHz0qPr7bDWuUS9W+tdb/XL87E3C6DrPDo/i3i8q4BH+s1jJpAJZALrTSA3DtabXNbt6gS4iYAQ6rc0aDDRYqd3gRXxI1w98LxnEwE1TjuaT4Q7DWg7vsMiH4twXHNjDC+9HiO+wx2GefT4dvNej0bkpQfv9d2jjTlF/CiDxHdeAm7R5DDM3OEOa12cuVqHRb1dH/cMOU1X63itmNOLfA/VrOcY9Xb4kN5DamvPfDz2hoHz6HKInhtw84MO+ezkMRPIBDYigdw42IhUU3NXJYCNA4xldyNEC53eBVbEj3D3ZIyxiQBdNyeHgRv5i/gR7nQchp7RiPhRzx58lbTHmH/P3NEv4o+RS6Qd4dH8E9/eCbgF1hAMabgF2hBNV+t6uGeip9ZxWzHnp7XW8aIc3Rxdb4e5Pg5rrYWX1nrnezM2C9C3dT7k4a4Cbho434llAplAJjBGArlxMEaKqZEJSAK8G4EbCj0LpogLebc4ivgRLjbnTns2EiJt5y/yDdzpjKERaQN3w/kAr9dLxHf6EbcXd9qLvG/U/Dfad49+xHVzT2z7JOAWe0MwzJyLLk1hiKardT20n5631jteK+b8uFr4crjDnKbOi+eO57ChvZ1Hh0W96ZfH1s0C8J2mw1r9LOLxLQg80m8eM4FMIBPYqARy42Cjkk3dTGCSADcREAbOexd6Y/AjjegJ6tlEgIbTjxZvEe40oO34Dot8LMJxrR6RD/BcX4dFXOCRvtNx2Fja0HGjxx/qezxG3I3sucijm39iq5eAWzgNwVoXcUhi7D4uXdej1aOrjXwP0XR9nJ6bHzDHdZjr47AhtZEf5711w8D5cZibS+THcamJjYLcLHDPWGKZQCaw0QnkxsFGJ5z6mcDZBLBxoP/1LpjG4kc67okaYxMBum7R6DBwI38R3+GRRoSjrxsR3/VEvcMdBu4Y2lHPXm3ouOF0ovmMhbue8Ob0HRZxF+G4lmN1EnCLpiEYZsZFl85yiKardT20n5631jteK+b8tNY6Hvw7TZ0Xzx3PYa6Pw6Lejusw15te9Thks6DHo/OzyDc3CnhUz3meCWQCmcBmJZAbB5uVdPbJBCQBbCBgLPtcBHDc4qhncQWNiB/hqHGjZyMh0nbzQS+Hj6EB7UgnwlFTj4jrfKO2F3f6vRoRv0e7njcfO41onpGPXrynZ+QlwiMvnG8eNzcBt2hyGFw53GGtizNX67Cot+vj0nOartbxWjH0HVvT6bn5Rb1dfet8xq6NfA/ZMGj16Hjw47IAl59ZkJsF0bOWeCaQCWx2ArlxsNmJZ79MwCSAjYT1biJArneB5fgOM1bXoM3eROidZ7QwjOYZ4WsTlpNFXNfXYZCL8Ejf8R02lrZMee60x98iL867w6CxkT0XecS1HBuXgFs0DcHg1C3Qhmi6WtfDpdRT67itmPPTWut4UY5ujq63w1wfh7la9HVch0X1tffN2CxAT+dnkW9sFHDToPacjzOBTCAT2MoEcuNgK9PP3pmASYCbCLiE894F01j8SMdYLj2bCKh32tGCMcKdBrQjvsMjjQiHvhsR3/VEvcMdBu4qacOPG85jNJ8xcNePvpy+w8Dvxdkjj8MScIumVgydHbd1ceZqWzH0dn2A18NpulrHa8Va9eBtiGY9Nzx2vR0G7pDeQ2rRux6tmwWoc/NxmPPYymMf3lXAY+07H2cCmUAmsAoJ5MbBKjwL6SETCBJofUtDtACKFli9/EjH2R5jEwG6kUeHR/4cd5G203GYmzexiN/rxfE3Uhv+nb7zwbm6o9MAz+k4LOIuwnt6Rjq9XqCToy8Bt8AagvUszsbu42buemyEx1ZN58dhmIvTdHN0PIe5Pg4bUtvju3XDwPlxmJtL5MdxqYmNgtwscF9piWUCmcAqJpAbB6v4rKSnTCBIgHcjcEOhd8HUw4+4ER5Y7robIdLuXdQ5nTE0MEenHc094vZ6ifg9+pFGhPdob+T8I38R3us70nG4w6K5Jz5LwC2ahmBQ5aJLMx6i6WpdD+2n5631jteKOT+ttfDquE5T58Vzx3OY69HT29U7zPWmVz0O2SyAjuvT6mcRjxsFPKrnPM8EMoFMYJUTyI2DVX520lsmsCABbiKAgvPeBdNY/EjHWR/jboRo8RbhkT/HdxjmEWlEuJv7Ih3X12HQiPDIS8R3uMN6fYPvxhj+oBt5dPhW9HRz322YWzQhA4e3Yq2LOKfX09v1cc+f6+NqHa8VQ9+xNZ2em1/U29W3zmfs2sj3kA2DVo9RPi4LaubdBdEzlngmkAlslwRy42C7PFPpMxNYkADvQFj2AYtucQXZ3gVWxI9wZ32MTQToujk5DNzIX8SPcKfjMPSMRsSPevbgq6Q9xvx75o5+EX+MXCLtCI/mvxNxt2gagiEjLro0ryGartb10H48d7W45uodtxUbWy/yyHnp0fV22JC5oN/QevWM81XeLMgPOqyfrXycCWQC2zWB3DjYrs9c+s4EFiTAuxG4odCzYIq4aOcWRxE/wiPbPRsJkbbzF/kG7nTG0Ii0gbvhfIDX6yXiO/2I24s77UXeN2r+G+27Rz/iurlvd6x1Aeh4mLvDhyxUnZ7D0Nv1AV4PV+9qHa8Vc3rw0VrveJFmPb+I53DXx2Fj19ae8bh1swBc58dhQ+bCPnwLAo/Ac2QCmUAmsBMSyI2DnfAs5hwygQUJcBMBFJz3LvTG4Ecake2eTQRoOP1o8RbhTgPaju+wyMciHNfqEfkAz/V1WMQFHuk7HYeNpQ0dN3r8ob7HY8TdyJ6LPLr5bxfMLbCGYG4RhyyGaLraqE+de2ut4/X4dn6cZiuG3k6znl/Ec7WtvVtr0dtpunrnu3XDwOk5zHmJ8nFcauZbEdyzlVgmkAnspARy42AnPZs5l0xgSQK8A4FvaehdMI3Fj3Sc/TE2EaDrFo0OAzfyF/EdHmlEOPq6EfFdT9Q73GHgjqEd9ezVho4bTieaz1i46wlvTt9hEXcRjmurPtyiaQiG+XLRpXMfoulqXQ/tp+et9Y7Xijk/rbXw6rhOU+fFc8dzmOvR09vVO8z1plc9DtksgI7r0+pnEY93FfConvM8E8gEMoGdlkBuHOy0ZzTnkwk0JlBvIqCsZ8HUw12kHelE0+jZSIi0exZ7Y2iMNf+xvETzd/oRtxfv0Y6ee6cBbo+XHu6i561Xx/EdFs19q3C3aIIXh7dirYs4p9fT2/VxObo+rtbxHBZ5HKLp+jg9Nz9gjusw18dhY9dGvodsGLR6jPJZNO+8uyB6xhLPBDKBnZxAbhzs5Gc355YJdCSw09/SMMaiE3H26EQLw0gjwt3TuIjr+joMuhEe6Tu+w8bSdnMH1uNvkRfn3WEb3XORR1zb7OEWTUMw+G9dyA3p43q47FyPjfDo/LjerVjk0c3R9XbY0N6t9a638z32ZgF6OI/OzyIe7yrg0XlPLBPIBDKBnZxAbhzs5Gc355YJrDOBVdhEgPVoceim1XMnQqQdLRgjPPIX8R0eaUS4m3s0H+CuZ4RH3MhLxHe4w9bjGzVuOI9RzzFw14++nL7DwO/F2WMjjm7R5DD0drjDWhdnrtZhUW/Xx2UEzc9+9qXycz93eO3yZz/7RPkn/+TqtcdRD+fHYZEXx23FIs0505MHEc/hQ3oPqa0943HrZgG4bi4Ocx5beezDv4iQmwVIJEcmkAns9gT27/YAcv6ZQCZwfgJ4sXTppZdOLyx6S4O+MNMFkL440wVWLz/SOd/x/AvPlk0Ep63+0INzinCngTrlUyPCVQMc5qU4MVyPRsTv8aJc9KH3zdJGT85VvdAHrkfDeVQN1FFHcWK43oNrP9TSd6Sj2uCwr+LEIg3gGzHUA/WHYHU2Y2g6P1Ef9uPR1X7gA389ubx3Srnxxosmx9nGgeO2Ys5Pa63jwZzTnJqu/nE8h7k+DmuthY3W+sry9GHrhoHz4zDnBY1aueTlWxHcs5VYJpAJ7PYEcuNgt38F5PwzgSUJYBMBQzcS+OGKwLlgql+wcRHEF2LgYvTwyUWd6iiOa/WoX4wu20iItHVOnA96OVw1wKFHx400gKsONRQDhzjO3VC+cnu9OL5qozf1lQuceSlODNcjXPWXaUPHDacBnuupGDj0qDixSAN4T89IR3uCw76KE8P1IUM1qTMEg4ZmMIam8+N6sFd9jOv3zFFPn94z97WBi67WYc6P4zks6uM05wyffeB4Dhva29U7zPV2vuufz45DzGk6rNXPIh7vKuCRHvKYCWQCmUAmMEsgNw7yKyETyAS6EqjvRsAmAgbuTOBCD4/1BZoudvRF3zK+cqFJfoSD44a+UO3ZRIAWe0bziXD16DSgzVx6NFDntIG7oVxcX6+XXo+Orxi8rHf+qkMN6Lmh8+fcwYs0HK4YatkzwsfoGXmMeoLfMup61AzBdK7s7/Q2og/76dH1dh6xWTCZuZbaHEBo1XS8Vsx5nDMnDxzXYUN6D6kVq3On+nN47kL1wM3FYc4jpFq55OXdBdUTkA8zgUwgEwgSyI2DIJiEM4FMYHkC9SYCKpbdjcBFF7h84YbzlkVdxI9w6NZDX7wu20RArdOuX7ByTooTizSAO75i4FBHfQBnXooTw/VoRHzty56KQW8Z3qMNPdUfWxv69VB/uMa81Adw54UYrit/GT6k5xAvqOVQv2Ng0KjnBWzsPq4H+rjherv6Ge/8DYO6vn6MnrHevCNX67BIc15t9sj1dpjr47Cot+M6zPV2vvXnrbtOzOk5DPxWP4t4vKuAR/rIYyaQCWQCmUCcQG4cxNnklUwgE+hIYMy3NHBBh/b1iz8u1PRFpfIj3E2lflG7bCMh0laP9KcYehNXDeD0rnxycT3CVYcaiqk2zt1QPjXAi3r24KoNTer3aEReWrRRqznicT1Uh/7AcR4VA4faihOLNID39Ix0tCc47FvjuKYjuu5wh6l36jpeKwYNx3V92E+PrbWOpzo8d7zIi+O2YpEmffAY8Rw+pPeQWnqtj/XP1vo6H7u5OMx5dDzoRlz8jsJ/uVnA9POYCWQCmUBfArlx0JdXsjOBTKAxgfpuhJ63NNQvCJct6iK+4qoRTUFf7PZsIkCP+vqilQs6XI9w51G5qKWO4sRw3WkswnGtHpGG9kQN+ypODNcj3Okrd2ztRV5wrR7OHzjqcdk8lYta8hUnhutj9ISO6uNxy3A1DoOW+qS24w7BXA/20qPrEdU67gybv+PgzJnzP+PAacZ66jB+PpzmfOXskeM5rNVPay26O01X73zrz093nZjTc5jzAo1WLnnYKOCmAT3kMRPIBDKBTKA/gdw46M8sKzKBTKAzgXoTAeV8SwMX3MD0heKQBRZfMEKT+oopjnM39EXwsk0E1Ks+e+p8wOGcFCcWaQB3fMXAoY7zgesRjmv1UC6ucT44177sqRg4DieG66o/hrb27NFGnRuqgev0qPPU+fTgyoU2dVp6Kr/WwbX1jEin9gNtxx2CQdP1AV4P18fVOp7DsFkwmdFcm1Y9FDlNhznNuaZnHziew4b2dh4dFvWuvevPyfpa/dhpOqzVzyIe7yrgsfaSjzOBTCATyAT6E8iNg/7MsiITyAQGJIBNBAz8lYbWP/UIvltgcUGH6/oiklzg+sJU+RGOmnrUL46XbSRE2s6jYuhL7z0aqFMdpwEO56/aiuPcDeVTAzzXM8KVC84yj8onN9KOcPUNDr1H2uC4oTpOAzX0qNoRTi6uK19x17Pm4/GYQ3tSV/0Rw9HhrZjro9o8d3qu1vEcds73/IYB8FrX1bditRbn446O67AhvVtrXQ7OM7D6Z2LEc3NxmPMIzVYuedgoyM2C6NlIPBPIBDKBYQnkxsGw/LI6E8gEBiSgmwiQwUbCGG9pqF+EckHGF5e0zEWg4sTIcUd90TzmJgJ6qfdlvpWLWvIVJ4br0TwjHDX1iLjaEzXsqzgxXI9wp6/czdJGHzecP/DU47J5Khe15CtODNe1Jx5v9FAf2svhrVjPHJymq3e8VuzMGZ3Z7Ly11vGg4Dye38XzXK3r47Cot+M6zPV2vvXnnrtOLNJzeKufRTxuFPBIH3nMBDKBTCATGDeB3DgYN89UywQygQEJ1G9p6NlEQFt9YaobAPqiM1qQka8a0CSOczf0xXTPJgK0qB35i3D1SA3oOb5i4HD+kUaEo7YeysW19XpRj/QHPdXfSm14cUP94To9RvPpwZXrem8U1tI34ji8zijy3VrreK0YesNPzccmQo3Vj1HrsNb5sTeOOly96+OwsWvVl57rzzjF63Pnx2Gtc4F+xMXvCvyXmwX1s5CPM4FMIBPYuARy42Djsk3lTCATGJBAvYkAKX4uAs6XLdL0BSu5qNMXoi2L1EgHWvXQF9jLNhFQ67TVHzj0GOFOA3XKp0aEqwY4zEtxYrgejYjf40W56EPvm6WNnpyreqEPXI+G86gaqKOO4sRwXXE83uzR29/xNYdF/ntqHbcVc35Onz7fWaseKp3m+Yqe52pbe7ta9B1aX3vXn2X1NX3s/DjM+YNOK5c8bBRw00B95HkmkAlkApnAxieQGwcbn3F2yAQygYEJ4IUiBj4XAYNvaeBnJLiFHnhckPFFJzCMHj65qFMdxXGtHvUL72UbCZG2vuDmfNDL4aoBDj06bqQBXHWooRg4xHHuhvKV2+vF8VUbvamvXODMS3FiuB7hqr9MGzpuOA3wXE/FnNZmYkO86JwXeXY9XK3jtWKL9GYfjKgOz78LwfVxmqrCc8dzGPiuj8NcveM5zNXSqx7rn1l6rT53mg5r9eN46AlN3lXAY+0lH2cCmUAmkAlsTgK5cbA5OWeXTCATGDGB+m6EjXxLQ/1imAvJCI+mqS/KezYRoMee+uJ67AVwpK3zpA94inBcq4dycY062hM456Q4MVyPcNWndsRXDXCorzgxXO/RBt+NFg1Xt1WY+h3Lg+ZLTdfH8cB3uMPaNfdOVPHfueH0cNVpnqs6d+Z4DnN9HDaktse3/mw6N5vzz5wfh7m5RH4cl5rYKMjNgvOfh0QygUwgE9iqBHLjYKuSz76ZQCYwSgL1JgJE8y0Ns2jXuwCuX8xThy/o+cRxka44MXLcMeJrX/ZUDFrL8B5t6Kn+2NrQr4f6q6/txMeaL+fnMnC8VqxXr9bFHQg15jTpX4+O57BanxoOb61vrWUvPQ7ZLIBOq8deHjcKeFTPeZ4JZAKZQCawtQnkxsHW5p/dM4FMYMQExnxLgy6A6xfoXGDqi2LlR7ibav0CvuduBO2pHulPMfQmrv6AU0f55OJ6hKsONRRTbZy7oXxqgBf17MFVG5rU79GIvLRoo1ZzxOPtNDSnob7rvKDn9Fsx1A/RnL1dYf7PMjo99HHDcR3WOp+xa51nYPXPm4jn/DjMzQ+arVzy8u6C6JlIPBPIBDKB1UkgNw5W57lIJ5lAJjByAvXdCD1vaeALWlriohOP9cUyF4YRX3HVoG591Bf2PZsI0KG+84frEe48Khe1nKfixHDdaSzCca0ekYb2RA37Kk4M1yPc6St3bO3aCx5vt6GZDfVeZ6167prDnB/HW4zNbxhMvmLmvn7Vl5673g5b3PucoqvF1aH15zrMzvRnSn1NHzs/DkON8+i4i3jYLMDPaG74qpc8zwQygUwgE1i9BHLjYPWek3SUCWQCG5BAvYmAFnxLAxfcwPSFri5G9UXxevmqgV6qg8f10Bf8yzYRUKv61Nb5gMM5KU4s0gDu+IqBQx3nA9cjHNfqoVxc43xwrn3ZUzFwHE4M11V/DG3tqdrAd8Ko8x0yp0Va7lqUp+O2YvgTjC0j6u3w1t5j17p56M8Od10x58dhbn6tPPQDl29B4FF95HkmkAlkApnAaieQGwer/fyku0wgE9iABPj/cOGvNPAvM3ATAe24kKxfKHPhqS+WyUWd8skFHvEjHDX1qBcCyzYSIm3nUTH0pfceDdSpjtMAh3mptuI4d0P51ADP9Yxw5YKzzKPyyY20axyPd+LQTIbMr0VHn3P2cnWtGDRmmvMfiug2EVxvh7X2bq2FR6fp6sGtR/1zor7Ox07PYc4LNFq55OVbEZh8HjOBTCAT2L4J7Jm8GGrcd9++k0znmUAmkAm0JsA/9Qg+znWRqhrRj84x+JGG9tfzZZsIyo20o/lEeI9Or0akrfPg+SKu6+swaEV4pB/x6WunH6MFZc+8uahcVuN6OQw6Dm/t43gOcz16ert6h7neLqvWzQLUOk2HtfpZxONdBTw674llAplAJpAJbJ8EcuNg+zxX6TQTyAQ2OYFV2ETAlKPFq4ujZxMh0o4WxREe+Yv4Do80ItzNPZoPcNczwiNurxfo7/ThFp29c3aLz0jDcR3W48txHeb6OGzs2iiL1g0D58dhbi7o3colL+8uiJ6xxDOBTCAT2N4J5MbB9n7+0n0mkAlsUgKL3tKgFnoXnb38nsXrGJsImFvk0eGRP8ddpO10HKbZ1+cRv9dLxK/7jf048s8+XKjx8VYeo0Vnj6cejYjbmonjOcz1cRjmOaTe1brsxt4sQA83H+dnEY93FfDovCeWCWQCmUAmsL0TyI2D7f38pftMIBPYogR4NwI3FKJFXrTo7OFH3AiPIunZSIi0o/lEuNOJuBHuNDDHCHfzj7hRzwh32uvBNlpfPbkFn14f+3xov956t8h1c4p4DnceHDZ2rfMNbOwNg9a5oHfE5V9EyM0CpJQjE8gEMoGdn0BuHOz85zhnmAlkAhucADcR0AbnvYvUsfiRjpt+zyYC6p12tPiNcKcBbcd3WORjEY5rbvR4cfWtWDSP1vqN5LkF4Zj9huq7Rfl6/Dkdhzm/DmuthdfWejevzdgsQN/W+ZCXb0Vwz1ZimUAmkAns/ARy42DnP8c5w0wgE9jEBHgHgvsrDWojWlD2LmgjfoSrB56PsYkALTcnh4Eb+Yv4Ee50HIae0ejlRzrAI5+LalbpmlvojuWPC8+x9CKdqE+Euzk7zNU7nsNcrfPfulmAWqfpsFY/i3i8q4BH5z2xTCATyAQygZ2dQG4c7OznN2eXCWQCW5wA70bghkK0SHULzoiLKfXwF+m4eHo2EiJt5y/yDdzpjKERaQMfOpznoZqrWu8WpJvttdfDIr5bJDvMaTiew1xtlFnrhoHTdJjzg96tXPLy7oLoGUs8E8gEMoHdl0BuHOy+5zxnnAlkAluUADcR0B7n0cIzWjCPwY80okh6NhGg4fSj+US404C24zss8rEIx7Ucm58AF6i9nXvqokU0ekbXnL7jOszVuvmNvVkQzcf5WeSbdxXw6LwnlglkAplAJrD7EsiNg933nOeMM4FMYAUS4B0I+ZaG2ZPRuwEQ8R0ebUS0fBkMqW3R3ykctzjtndt6NNwCOOq7jOv6uxqHudrIx9gbBj1+Im5+0GH0bCWeCWQCmUAmwARy44BJ5DETyAQygS1KoHUTAfZ6FsaOC41oMRzhqHGj526ESDvy6PAxNNw8aizqU/PGfrwRfXsWtGPOZ2hft8Bd5K+Hv8xbpOXwZVr0vBmbBejl/CzyjbsKuGlAr3nMBDKBTCATyARcArlx4FJJLBPIBDKBLUxgp7+lIVogu80CPA0R3quz6CmNeiyq6bkWee3R2EiuW3CO2c8tXlv1e2tb59Ki6zit+q2bBcjBaTqs1c8iHt+CwGPr85C8TCATyAQygd2dQG4c7O7nP2efCWQCK57AKmwiIKKehW/PnQiRdrSQj/Aef3zKIy1eX89xIzTX42PMGrcIHaq/Xk23mP7/2zvX3bhxGIwGTdo07/+wvSQtCg7ArVYlZVKSZ2T7BMhKlilejvqH32omXi7RGBG7TNyoYGD5tNa8/KK2aidCAWKB96+FdQhAAAIQ2CKAcLBFiPcQgAAEFiEQ/UiD17x6zXXW3vNjYZohIohfL0dv3cpF1jK5ez5kPRu35euo77yGNluPNrbZfZn4EdvePCTvEbFA9luxrZyzdioU6Cix+IEABCAAAQj0EEA46KHGHghAAAILENDbCCooeE2x1+Rm7D1bb93DkxESPN9ePV7MrL3lx8vFsr3ymtXYZnlYDfOWj+ieGflJLlGhQPO24lprVh2Wnfj1bPU7CxALlD4jBCAAAQjMIIBwMIMiPiAAAQg8mICKCJKGzL1G12uiZ9l7fiw8GRFB9md8e3VaeVhro/stn95api7PR2bda0QzPjK2VoO75/6Z9WUFgrIuKw9rzeMTtVU7EQpUNCjzYA4BCEAAAhCYQQDhYAZFfEAAAhBYiIDeQDjzn3q0cI804HsJBXv5terfY81rakdj9frVJnk0fr1/RCBQX63crHcWg6idxBRbvVWgo+bCCAEIQAACEJhNAOFgNlH8QQACEFiMgN5GUEHBa7CtJtezlRIz9i0/Fq7sbQTLR2stm0/Ll8WhZX+Wd1bj21ub1TD3+vL2zRAHxPdWrq33HjNrj2WrdiIUIBZ4J806BCAAAQjsQQDhYA+q+IQABCCwKAEVESQ9mXsNtNcMz7D3fGSQ7S0sWLnMyNvye7Y1bW7vXdcsYUDzjtTRsrEaf/Uto7XX2qN2KhToWPpiDgEIQAACENibAMLB3oTxDwEIQGBRAnoD4UgfacigfIS4kMlPbFcQI7QxzeZ+b/vZwoDmH6l/y8Zq+NW/jK393l7dw+2CkiRzCEAAAhB4FAGEg0eRJy4EIACBhQhERQRJ2bqN4DXAlq348OzlXfkTtSv39MyPIDL01LX6nr3EgLpubcLr9fo5Yuc1+upry0drv+4VsYAvOlSijBCAAAQgsAIBhIMVToEcIAABCCxG4N4faSjLz4gFGdsyxow5YsO/FO8lBPwbuf1/9Wt7bdDr9fK51eCr3Zaflo96rwoFfBRB6TJCAAIQgMBKBBAOVjoNcoEABCCwIIG9RISyVO9mQmmj86xYkLXXOI8e9xAmHtnYj/Csm+yIr+ieVnNfxtnyt+XH289HEUrKzCEAAQhAYFUCCAerngx5QQACEFiQQPQjDREhINPQR/wproxf3aPjyF71wRgn4DXTEQ+ZvVtNfRkv4nfLX8uH7H17e7uF5HZBSZ45BCAAAQisTADhYOXTITcIQAACixPQ2wgqKGw13lEBYMtPjSXqt9yXjVHuLeez/JQ+jzpvNcyZmnr8bDXzdfxIjIjPLT+lDxEMEAvqk+AZAhCAAASOQADh4AinRI4QgAAEDkBARQRJVYUETTvSXEeb/4gvjatj1Lfa12NPzNpHz/Necbea3Z5cI3tG4pYNeCSW2GTiRfxv+bN8cLsgelrYQQACEIDAygQQDlY+HXKDAAQgcFACKhyUf+rRKiXaGGca/6hPK59MHGu/tTaSj+Xv0WtbzXNPflbDHfWTzScSK+Kz5Ue/6FBFg2gt2EEAAhCAAARWJYBwsOrJkBcEIACBExEQIeHbt2+3mwiR5jzTbEf81Sgz/uu95XNP7HL/meetxjpTd6SJr/1lYkf8R/yJjQgFLy8vtz+lWOfEMwQgAAEIQODIBBAOjnx65A4BCEDggARURJDU9WaCzKNNeLbpj/qVHOqfbKx6f+Z5JM9MnC3bSJO85SP6PtK0e76yeUZjRfyWNnqrQEcvX9YhAAEIQAACRyaAcHDk0yN3CEAAAgcnoMKB3kawysk01L2NfiaGlWO51ptD6eOI82hjHqmtbMwj9mqTySEaw7MToQCxQMkzQgACEIDA2QkgHJz9hKkPAhCAwEEIRESEupSehn+0se+JWec98jySf6axHsmxtddrxFt7ynfZGqLxInYiFIjd169fy5SYQwACEIAABE5PAOHg9EdMgRCAAASOSaD8SMPHx8etiEzTnrGtCY0057UvfR7JR32sNEYa7Wy+WVFA/WdzidpLPvKdBfKLWKC0GSEAAQhA4IoEEA6ueOrUDAEIQOBgBCwRoS6hpzHv2VPH3UNkqGMc/blXECjrjjb7vXvqHEUoUNGg9MkcAhCAAAQgcEUCCAdXPHVqhgAEIHBgAnL7QBp++V4EvYmwVc6IQDCydyuvI4kOdWO9VVv2fY8wIDGy+1p16K0CHbM1YA8BCEAAAhA4KwGEg7OeLHVBAAIQuAgBEQ++f//+n4jQ04zPEgdm+TnL0WWbeq/uHj8tgaCOI0IBYkFNhWcIQAACEIDAXwIIB39ZMIMABCAAgYMTUBFByrBuI/SICorkHqLAPWJoPdGxp2mP+la7kRgZgUDiaazX19dbeAQDPQVGCEAAAhCAgE8A4cBnwxsIQAACEDgwARUOytsIW+WMCAul7xUFgDK/e8y1QZ8RKysOSEwvvggGiAUzTgUfEIAABCBwJQIIB1c6bWqFAAQgcGECIiT8+PHjSf/sYy+KWeLCVvxHiw9e472Vd/Z9jyhQxojkKWIBX3RYUmMOAQhAAAIQyBFAOMjxwhoCEIAABE5AQEUEKUVvJmhZMxv2e4kMmvtK46ggUNYSEQdKe5m/vb3dlr58+VK/4hkCEIAABCAAgSQBhIMkMMwhAAEIQOBcBFQ4kNsIOt+qcKa40Ir1SOFhZuPfqlHe9QgD6vP5+Vmnt1GEAsSC/yHhAQIQgAAEIDBMAOFgGCEOIAABCEDgLAR+/fp1+1OPGRGhVfu9BIZWDo98NyII1HnXAkH5XoUCHct3zCEAAQhAAAIQGCeAcDDOEA8QgAAEIHBSAiIk/Pz581adzOVHx9vDjv9ZSXSYKQB4yFrCgLeH2wUeGdYhAAEIQAACcwkgHMzliTcIQAACEDgpAUtEaJV6L4GhlcNK73qEASt/vVWgo2XDGgQgAAEIQAACcwkgHMzliTcIQAACELgAARUF5DaCzmeUPdPXjHw8H7NEAM9/vS7x5BexoCbDMwQgAAEIQOA+BBAO7sOZKBCAAAQgcGIC0vDPFhFOjCtcGh9FCKPCEAIQgAAEILArAYSDXfHiHAIQgAAErkZARQSp+yg3CFY6I71VoONKuZELBCAAAQhA4KoEEA6uevLUDQEIQAACuxNQ4YDbCD5q/djDy8vL0+fPn31D3kAAAhCAAAQg8DACCAcPQ09gCEAAAhC4GgG9jaCCwhXqV2HAqlXEAvlBMLDosAYBCEAAAhBYhwDCwTpnQSYQgAAEIHAhAr9//356f3+/VSzz1s8qQkNLBGjlX7779OnTk/wiFpRUmEMAAhCAAATWJvAH8TnTu33pHFkAAAAASUVORK5CYII="
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Vectors are drawn as arrows with a starting point and an ending point. The black arrow above is the vector $\\begin{bmatrix}2 \\\\\\ 1\\end{bmatrix}$ with a starting point at the origin $(0, 0)$. Since $x = 2, y = 1$, we move 2 units in the positive X-direction and 1 unit in the positive Y-direction to arrive at the endpoint. Vectors are generally drawn as an arrow to differentiate them from a line segment.\n",
    "\n",
    "Vectors are not limited to just 2 components. This same process scales to vectors with any number of components. For example, here is a vector with 3 components:\n",
    "\n",
    "$$\\begin{bmatrix}1 \\\\ 2 \\\\ 3\\end{bmatrix}$$\n",
    "\n",
    "This represents a vector in 3D. Geometrically, we allow $x = 1, y = 2, z = 3$. In order to visualize this vector, we need a 3D coordinate system:\n",
    "\n",
    "<span style=\"display: none\">(attachment:image.png)</span>\n",
    "<img src=\"attachment:image.png\" width=\"500\">\n",
    "<figcaption style=\"display: block; text-align:center;\">(<a href=\"https://www.geogebra.org/3d/nhytbgup\">image source</a>)</figcaption>\n",
    "\n",
    "In order to visualize this vector, we move 1 unit in X-direction (red axis), 2 units in the Y-direction (green axis), and 3 in the Z-direction (blue axis). Note, the light blue box in the plot above is just a visualization aid.\n",
    "\n",
    "Beyond 3 components, it becomes difficult (or impossible) to visualize, but the vectors are still completely valid. For example:\n",
    "\n",
    "$$\\begin{bmatrix}1 \\\\ 0 \\\\ 2 \\\\ 1 \\\\ 4\\end{bmatrix}$$\n",
    "\n",
    "This vector represents a vector in 5D. Good luck trying to visualize that."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Magnitude and direction\n",
    "\n",
    "Another way to reason about a vector is by viewing it as something with **magnitude** (how long it is) and **direction** (the direction it is pointing in). For example, when visualizing the following vector:\n",
    "\n",
    "$$\\begin{bmatrix}2 \\\\ 0\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(6, 6))\n",
    "axhline(y=0, color='k', linewidth=0.5)\n",
    "axvline(x=0, color='k', linewidth=0.5)\n",
    "grid(linestyle='dotted')\n",
    "axis([-1, 3, -1, 3])\n",
    "xticks([-1, 0, 1, 2, 3])\n",
    "yticks([-1, 0, 1, 2, 3])\n",
    "quiver([0], [0], 2, 0, angles='xy', scale_units='xy', scale=1, color='b')\n",
    "annotate('Vector [2, 0]', xy=(1, 0.25), ha='center', color='blue')\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can clearly see that the vector has a length of 2 (because $x = 2$), and is pointing due east. Therefore, the magnitude of this vector is $2$ and the direction is east. It's important to note that the components of a vector are what determine the magnitude and direction. The idea behind magnitude and direction become clear when we visualize vectors with different components. For example, if we visualize the following two vectors:\n",
    "\n",
    "$$\\begin{bmatrix}2 \\\\ 0\\end{bmatrix}, \\begin{bmatrix}0 \\\\ 2\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(6, 6))\n",
    "axhline(y=0, color='k', linewidth=0.5)\n",
    "axvline(x=0, color='k', linewidth=0.5)\n",
    "grid(linestyle='dotted')\n",
    "axis([-1, 3, -1, 3])\n",
    "xticks([-1, 0, 1, 2, 3])\n",
    "yticks([-1, 0, 1, 2, 3])\n",
    "quiver([0], [0], 2, 0, angles='xy', scale_units='xy', scale=1, color='b')\n",
    "quiver([0], [0], 0, 2, angles='xy', scale_units='xy', scale=1, color='r')\n",
    "annotate('Vector [2, 0]', xy=(1, 0.15), ha='center', color='blue')\n",
    "annotate('Vector [0, 2]', xy=(0.1, 1), ha='left', color='red')\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The vectors are clearly pointing in different directions even though they have the same length. Because these vectors point in a different _direction_, they are completely different vectors. Similarly, vectors can point in the same direction, but have a different _magnitude_ (i.e., length). Consider the following 2 vectors:\n",
    "\n",
    "$$\\begin{bmatrix}2 \\\\ 2\\end{bmatrix}, \\begin{bmatrix}4 \\\\ 4\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(6, 6))\n",
    "axis([0, 5, 0, 5])\n",
    "grid(linestyle='dotted')\n",
    "annotate('Vector [2, 2]', xy=(2, 1.6), ha='left', color='red')\n",
    "annotate('Vector [4, 4]', xy=(4, 3.6), ha='left', color='blue')\n",
    "quiver([0], [0], 4, 4, angles='xy', scale_units='xy', scale=1, color='blue')\n",
    "quiver([0], [0], 2, 2, angles='xy', scale_units='xy', scale=1, color='red')\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "They are pointing in the same direction, but the blue vector is twice as long as the red vector, thus the 2 vectors have different magnitudes. These are 2 different vectors.\n",
    "\n",
    "An important thing to call out is that magnitude can be different without affecting direction and vice versa as was demonstrated above. Or, they can both be different:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(6, 6))\n",
    "axis([0, 3, 0, 3])\n",
    "grid(linestyle='dotted')\n",
    "xticks([0, 1, 2, 3])\n",
    "yticks([0, 1, 2, 3])\n",
    "annotate('Vector [1, 1]', xy=(1, 1), xytext=(1, 1), ha='left', color='red')\n",
    "annotate('Vector [2, 1]', xy=(2, 1), xytext=(2, 1), ha='left', color='blue')\n",
    "quiver([0], [0], 1, 1, angles='xy', scale_units='xy', scale=1, color='red')\n",
    "quiver([0], [0], 2, 1, angles='xy', scale_units='xy', scale=1, color='blue')\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These 2 vectors are pointing in different directions _and_ have different lengths."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Representing vectors\n",
    "\n",
    "Now that we have an idea of what a vector is, let's see how we can represent it. In algebra, we can allow a symbol to represent a value:\n",
    "\n",
    "$$x = 5$$\n",
    "\n",
    "In this case, $x$ represents the number $5$. Similarly, we can assign vectors to symbols. The symbol names can be whatever you want, but conventionally, symbols that represent vectors tend to have an arrow over the top of them to differentiate them from the standard algebraic variable above:\n",
    "\n",
    "$$\\vec{a} = \\begin{bmatrix}3 \\\\ 5\\end{bmatrix}$$\n",
    "\n",
    "In some instances, you may see a vector symbol in bold:\n",
    "\n",
    "$$\\textbf{a} = \\begin{bmatrix}3 \\\\ 5\\end{bmatrix}$$\n",
    "\n",
    "When referencing a vector, I'll stick to the arrow-based method."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Okay, now that we understand how a vector is represented with a symbol, let's see how we can represent a vector in Python. Let's code up the vector:\n",
    "\n",
    "$$\\vec{v} = \\begin{bmatrix}1 \\\\ 2\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "hideCode": false
   },
   "outputs": [],
   "source": [
    "v = [1, 2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Fortunately, since a vector is just a list of numbers, a Python list is perfect for representing a vector. In the future, we may choose to represent the vector as a class, but for now, let's keep things simple."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "hideCode": false,
    "hidePrompt": false
   },
   "source": [
    "## Operating on vectors\n",
    "\n",
    "Vectors wouldn't be all that interesting if we couldn't do anything with them. Fortunately, vectors behave a lot like regular numbers in that you can use many of the same arithmetic operations we learned in grade school with vectors.\n",
    "\n",
    "### Vector addition\n",
    "\n",
    "One of the first operations we all learned was that of simple addition. Given 2 numbers, addition produces a new number that is the result of combining the two numbers on the left:\n",
    "\n",
    "$$3 + 4 = 7$$\n",
    "\n",
    "The same operation applies to vectors. The result of adding 2 vectors is another vector:\n",
    "\n",
    "$$\\begin{bmatrix}1 \\\\ 2\\end{bmatrix} + \\begin{bmatrix}3 \\\\ 1\\end{bmatrix} = \\begin{bmatrix}1 + 3 \\\\ 2 + 1\\end{bmatrix} = \\begin{bmatrix}4 \\\\ 3\\end{bmatrix}$$\n",
    "\n",
    "The addition of vectors is done _component wise_, that is the first component of $\\begin{bmatrix}1 \\\\\\ 2\\end{bmatrix}$  (which is $1$) is added to the first component of $\\begin{bmatrix}3 \\\\\\ 1\\end{bmatrix}$ (which is 3) to produce the first component of the result (which is $1 + 3 = 4$). This process is repeated for all the components. Since both operands are vectors, the result is also a vector:\n",
    "\n",
    "$$\\begin{bmatrix}4 \\\\ 3\\end{bmatrix}$$\n",
    "\n",
    "Just as simple addition can be visualized by using a number line, we can visualize vector addition. Following the steps from above, we arrive at a resulting vector of $\\begin{bmatrix}4 \\\\\\ 3\\end{bmatrix}$. Below, the red arrow represents the vector $\\begin{bmatrix}1 \\\\\\ 2\\end{bmatrix}$ and the blue arrow represents the vector $\\begin{bmatrix}3 \\\\\\ 1\\end{bmatrix}$. The magenta arrow represents the vector $\\begin{bmatrix}4 \\\\\\ 3\\end{bmatrix}$ which is the result of adding the the 2 vectors."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(6, 6))\n",
    "axis([0, 5, 0, 5])\n",
    "grid(linestyle='dotted')\n",
    "quiver([0], [0], 1, 2, angles='xy', scale_units='xy', scale=1, color='red')\n",
    "quiver([1], [2], 3, 1, angles='xy', scale_units='xy', scale=1, color='blue')\n",
    "quiver([0], [0], 4, 3, angles='xy', scale_units='xy', scale=1, color='magenta')\n",
    "\n",
    "v1 = mpatches.Patch(color='red', label='Vector [1, 2]')\n",
    "v2 = mpatches.Patch(color='blue', label='Vector [3, 1]')\n",
    "v3 = mpatches.Patch(color='magenta', label='Vector [4, 3]')\n",
    "legend(handles=[v1, v2, v3])\n",
    "\n",
    "annotate('[1, 2] + [3, 1] = [4, 3]', xy=(2.3, 1.5), ha='left', color='magenta')\n",
    "\n",
    "show()"
   ]
  },
  {
   "attachments": {
    "image.png": {
     "image/png": 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"
    }
   },
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Adding 2 scalar numbers can be visualized using a number line. To add the numbers $3$ and $4$:\n",
    "\n",
    "<span style=\"display: none\">(attachment:image.png)</span>\n",
    "<img src=\"attachment:image.png\" width=\"300\" alt=\"\">\n",
    "<figcaption style=\"display: block; text-align:center;\">(<a href=\"http://mathandmultimedia.com/2015/07/25/add-integers-using-the-number-line/\">image source</a>)</figcaption>\n",
    "\n",
    "We simply move $3$ units in the positive direction and then from that spot move $4$ more units in the positive direction. You can also imagine this as if we had 2 ropes of length $3$ and $4$ and we lined them up end to end, then the combined length of the 2 ropes would be $7$.\n",
    "\n",
    "This analogy of addition is useful for understanding vector addition, because the addition of 2 vectors is conceptually the result of lining up the 2 vectors end-to-end to arrive at the result. In the above example, the vector $\\begin{bmatrix}1 \\\\\\ 2\\end{bmatrix}$ is plotted starting from the origin like we did above. Next, the vector $\\begin{bmatrix}3 \\\\\\ 1\\end{bmatrix}$ is plotted starting at where $\\begin{bmatrix}1 \\\\\\ 2\\end{bmatrix}$ ends. In total, after lining up the 2 vectors end-to-end, you can see that we moved 4 units to the right and 3 units up which is precisely the resulting vector $\\begin{bmatrix}4 \\\\\\ 3\\end{bmatrix}$.\n",
    "\n",
    "You can imagine the resulting vector (the magenta vector above) like a route—traveling 1 unit east and 2 units north followed by traveling 3 units east and 1 unit north, you arrive at the location $\\begin{bmatrix}4 \\\\\\ 3\\end{bmatrix}$. You would arrive at the same spot by simply starting from the origin and traveling to $\\begin{bmatrix}4 \\\\\\ 3\\end{bmatrix}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Commutability\n",
    "\n",
    "Like simple addition, vector addition is commutative. For example:\n",
    "\n",
    "$$\\begin{bmatrix}1 \\\\ 2\\end{bmatrix} + \\begin{bmatrix}3 \\\\ 1\\end{bmatrix} = \\begin{bmatrix}3 \\\\ 1\\end{bmatrix} + \\begin{bmatrix}1 \\\\ 2\\end{bmatrix} = \\begin{bmatrix}4 \\\\ 3\\end{bmatrix}$$\n",
    "\n",
    "We still arrive at the same result no matter which comes first. If we plot $\\begin{bmatrix}3 \\\\\\ 1\\end{bmatrix}$ first and then add $\\begin{bmatrix}1 \\\\\\ 2\\end{bmatrix}$, as expected, the addition still takes us to the same resulting vector:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(6, 6))\n",
    "axis([0, 5, 0, 5])\n",
    "grid(linestyle='dotted')\n",
    "quiver([0], [0], 3, 1, angles='xy', scale_units='xy', scale=1, color='blue')\n",
    "quiver([3], [1], 1, 2, angles='xy', scale_units='xy', scale=1, color='red')\n",
    "quiver([0], [0], 4, 3, angles='xy', scale_units='xy', scale=1, color='magenta')\n",
    "\n",
    "v1 = mpatches.Patch(color='blue', label='Vector [3, 1]')\n",
    "v2 = mpatches.Patch(color='red', label='Vector [1, 2]')\n",
    "v3 = mpatches.Patch(color='magenta', label='Vector [4, 3]')\n",
    "legend(handles=[v1, v2, v3])\n",
    "\n",
    "annotate('[3, 1] + [1, 2] = [4, 3]', xy=(0.5, 2), xytext=(0.5, 2), ha='left', color='magenta')\n",
    "\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Vector subtraction\n",
    "Unsurprisingly, subtraction is done component wise just like addition:\n",
    "\n",
    "$$\\begin{bmatrix}3 \\\\ 5\\end{bmatrix} - \\begin{bmatrix}2 \\\\ 2\\end{bmatrix} = \\begin{bmatrix}3 - 2 \\\\ 5 - 2\\end{bmatrix} = \\begin{bmatrix}1 \\\\ 3\\end{bmatrix}$$\n",
    "\n",
    "Another way to view subtraction is as negative addition. An intuitive reason for why this is useful can be seen when vector subtraction is visualized. For example, the above is equivalent to:\n",
    "\n",
    "$$\\begin{bmatrix}3 \\\\ 5\\end{bmatrix} + \\begin{bmatrix}-2 \\\\ -2\\end{bmatrix} = \\begin{bmatrix}3 + (-2) \\\\ 5 + (-2)\\end{bmatrix} = \\begin{bmatrix}1 \\\\ 3\\end{bmatrix}$$\n",
    "\n",
    "Using the route analogy, this is equivalent to traveling 3 units to the right, 5 units up (to get to $\\begin{bmatrix}3 \\\\\\ 5\\end{bmatrix}$) followed by 2 units to the left and 2 units down (equivalent to $\\begin{bmatrix}-2 \\\\\\ -2\\end{bmatrix}$). Below, the red vector is $\\begin{bmatrix}3 \\\\\\ 5\\end{bmatrix}$, the blue vector is $\\begin{bmatrix}-2 \\\\\\ -2\\end{bmatrix}$, and the magenta vector as the result $\\begin{bmatrix}1 \\\\\\ 3\\end{bmatrix}$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(6, 6))\n",
    "plot([3], [5], markersize=5, color='red')\n",
    "plot([2], [2], markersize=5, color='blue')\n",
    "axis([0, 5, 0, 6])\n",
    "grid(linestyle='dotted')\n",
    "quiver([0, 3, 0], [0, 5, 0], [3, -2, 1], [5, -2, 3], angles='xy', scale_units='xy', scale=1, color=['red', 'blue', 'magenta'])\n",
    "annotate('Vector [3, 5]', xy=(1.7, 2.5), xytext=(1.7, 2.5), ha='left', color='red')\n",
    "annotate('Vector [-2, -2]', xy=(1.2, 4.5), xytext=(1.2, 4.5), ha='left', color='blue')\n",
    "annotate('[1, 3]', xy=(0.2, 2), xytext=(0.2, 2), ha='left', color='magenta')\n",
    "\n",
    "v1 = mpatches.Patch(color='red', label='Vector [3, 5]')\n",
    "v2 = mpatches.Patch(color='blue', label='Vector [-2, -2]')\n",
    "v3 = mpatches.Patch(color='magenta', label='Vector [1, 3]')\n",
    "legend(handles=[v1, v2, v3])\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Scalar multiplication with a vector\n",
    "\n",
    "Scalar multiplication is performed by multiplying each component in the vector by a scalar number:\n",
    "\n",
    "$$3 * \\begin{bmatrix}1 \\\\ 2\\end{bmatrix} = \\begin{bmatrix}3 * 1 \\\\ 3 * 2\\end{bmatrix} = \\begin{bmatrix}3 \\\\ 6\\end{bmatrix}$$\n",
    "\n",
    "Conceptually, this is the same thing as repeated addition:\n",
    "\n",
    "$$3 * \\begin{bmatrix}1 \\\\ 2\\end{bmatrix} = \\begin{bmatrix}1 \\\\ 2\\end{bmatrix} + \\begin{bmatrix}1 \\\\ 2\\end{bmatrix} + \\begin{bmatrix}1 \\\\ 2\\end{bmatrix} = \\begin{bmatrix}3 \\\\ 6\\end{bmatrix}$$\n",
    "\n",
    "Geometrically, this can be thought of as stretching or shrinking the vector (depending on what the scalar number is). Below, the plot on the left is the result of multiplying each component by 3 and the plot on the right is the concept of repeated addition. Using what we know about addition from above, you can visualize scalar multiplication as lining up 3 copies of the vector $\\begin{bmatrix}1 \\\\\\ 2\\end{bmatrix}$ end-to-end. The key takeaway is that of the fact that the resulting vector is simply a scaled version of the original vector."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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sSh0DgBbqGLBBDqtZlToGAA3UMWAGDGQckDoGAC3UMWAGHFpzQOoYADRQx4AZMZDRSh0DgBbqGDAjDq9ppY4BQAN1DJghAxmN1DEAaKGOATPkEJtG6hgANFDHgBkzkLEfdQwAWqhjwIw5zGY/6hgANFDHgDkwkLEXdQwAWqhjwBw41GYv6hgANFDHgDkxkHETdQwAWqhjwJw43OYm6hgANFDHgDkykJFEHQOAVuoYMEcOuUmijgFAI3UMmDMDGeoYALRRx4A5c9iNOgYATdQxYAGmHshKKZtKKZ8qpbxjngtisdQxgI2xPw6YOgYswFoOvZ+b5OJ5LYRuqGMAG2Z/HCJ1DFiQqQayUsqdkzw2yavmuxwWSR0D2Bj744Cdd546BizEtIffr0jywiS72h5QStlWStleStm+Y8eOmSyO+VLHADbM/jhUD3948vrXJ8cfr44Bc7XqQFZKOSnJ5bXW8w/0uFrr6bXWLbXWLZs3b57ZApkPdQxgY+yPA/eBD0wq2fvfnzztaV2vBhiwaQ7BH5Tk8aWUryQ5O8kjSimvm+uqmDt1DGDD7I9D9fGPJ49/fHKPe0yul9LteoBBW3Ugq7W+qNZ651rrXZI8Ncm/11q9VNRj6hjAxtkfB+qii5ITT0yuuSY55piuVwOMgMPwEVLHAKDBJZckJ5yQXHHF5Pqxx3a7HmAU1vQO1VrrfyT5j7mshIVQxwBmz/44AF/7WvLIRybf+tbk+q1vnfz4j3e7JmAUHIqPjDoGAPu4/PJJGfvqV3ffduyx3jsGLISBbETUMQDYx5VXJo9+dPKFL+x9u/ePAQvicHxE1DEA2MO11yaPfWxy4YX73+f9Y8CCGMhGQh0DgD388IfJE5+YfOQjzfcrZMCCOCQfCXUMAFbs3Jls3Zq8733tjzGQAQtiIBsBdQwA9nD11cmppybnnZe88IX733+nOyW3uc3i1wWMksPyEVDHAGAPt7lN8pCHJA9+cPKud+2+/e53n/zu/WPAAhnIBk4dA4AW55yTXHTR5PJRRyWf+MRkKHO6IrBAa/rB0PSPOgYADXbtSk47bff1F784ue1tkze9KfnSl7pbFzA6BrIBU8cAoMW+dWzr1snl+943Oe647tYFjI7D8wFTxwCgQVMdO3iP16hLWfyagNEykA2UOgYALdrqGEAHHKIPlDoGAA1Wq2MAC2YgGyB1DABaqGPAknGYPkDqGAA0UMeAJWQgGxh1DABaqGPAEnKoPjDqGAA0UMeAJWUgGxB1DABaqGPAknK4PiDqGAA0UMeAJWYgGwh1DABaqGPAEnPIPhDqGAA0UMeAJWcgGwB1DABaqGPAknPYPgDqGAA0UMeAHjCQ9Zw6BgAt1DGgBxy695w6BgAN1DGgJwxkPaaOAUALdQzoCYfvPaaOAUADdQzoEQNZT6ljANBCHQN6xCF8T6ljANBAHQN6xkDWQ+oYALRQx4CecRjfQ+oYADRQx4AeMpD1jDoGAC3UMaCHHMr3jDoGAA3UMaCnDGQ9oo4BQAt1DOgph/M9oo4BQAN1DOgxA1lPqGMA0EIdA3rMIX1PqGMA0EAdA3rOQNYD6hgAtFDHgJ5zWN8D6hgANFDHgAEwkC05dQwAWqhjwAA4tF9y6hgANFDHgIEwkC0xdQwAWqhjwEA4vF9i6hgANFDHgAExkC0pdQwAWqhjwIA4xF9S6hgANFDHgIExkC0hdQwAWqhjwMCsephfSrlZKeUTpZRPl1I+W0o5bbU/w8aoYwD9YI9cMHUMGKBpvov9MMkjaq3XlFIOSfLhUsq7aq0fm/PaRkkdA+gVe+QiqWPAAK06kNVaa5JrVq4esvKrznNRY6aOAfSHPXKB1DFgoKZqL6WUTaWUC5NcnuR9tdaPNzxmWylleyll+44dO2a9zlFQxwD6Z7U90v44I+oYMFBTHe7XWm+otd4nyZ2T3K+UckzDY06vtW6ptW7ZvHnzrNc5CuoYQP+stkfaH2dAHQMGbE39pdZ6ZZIPJHnMfJYzXuoYQL/ZI+dIHQMGbJpPWdxcSrn1yuWbJzkhyefnvbCxUccA+sceuQDqGDBw03xHu2OS15ZSNmUywL2p1vqO+S5rXNQxgN6yR86bOgYM3DSfsvifSY5bwFpGSx0D6Cd75JypY8AI6DAdU8cAoIU6BoyAQ/+OqWMA0EAdA0bCQNYhdQwAWqhjwEg4/O+QOgYADdQxYEQMZB1RxwCghToGjIgRoCPqGAA0UMeAkTGQdUAdA4AW6hgwMsaADqhjANBAHQNGyEC2YOoYALRQx4ARMgosmDoGAA3UMWCkDGQLpI4BQAt1DBgp48ACqWMA0EAdA0bMQLYg6hgAtFDHgBEzEiyIOgYADdQxYOQMZAugjgFAC3UMGDljwQKoYwDQQB0DMJDNmzoGAC3UMQAD2bypYwDQQB0DSGIgmyt1DABaqGMASQxkc6WOAUADdQzgJgayOVHHAKCFOgZwEyPCnKhjANBAHQPYi4FsDtQxAGihjgHsxZgwB+oYADRQxwD2YyCbMXUMAFqoYwD7MSrMmDoGAA3UMYBGBrIZUscAoIU6BtDIuDBD6hgANFDHAFoZyGZEHQOAFuoYQCsjw4yoYwDQQB0DOCAD2QyoYwDQQh0DOCBjwwyoYwDQQB0DWJWBbIPUMQBooY4BrMrosEHqGAA0UMcApmIg2wB1DABaqGMAUzE+bIA6BgAN1DGAqRnI1kkdA4AW6hjA1IwQ66SOAUADdQxgTQxk66COAUALdQxgTYwR66COAUADdQxgzQxka6SOAUALdQxgzYwSa6SOAUADdQxgXQxka6COAUALdQxgXYwTa6COAUADdQxg3QxkU1LHAKCFOgawbquOFKWUI0opHyilfK6U8tlSynMXsbBlo44BsCf74wp1DGBDpvmOuTPJ82utF5RSDk9yfinlfbXWz815bUtDHQOgwej3xyTqGMAGrTpW1Fq/WWu9YOXy1UkuTnKneS9smahjAOzL/rji859PNm2aXFbHANZsTd81Syl3SXJcko833LctybYkOfLII2ewtOVwYx07OefmX/MEdQyA/Yxxf7zJb/1Wco97JBdcoI4BrMPUo0Up5ZZJ3prk1FrrVfveX2s9vda6pda6ZfPmzbNcY6fOOis5/sv/lGfkn9UxAPYz1v0xSfK97yWPeUxyxBHJS1+qjgGsw1QDWSnlkEw2m7NqrW+b75KWx86dyade+Ib8Q347n8kx6hgAexnr/pgkue665HGPS84/P7nXvbpeDUBvTfMpiyXJGUkurrX+1fyXtDzOe8H/zv+6/Ok5KDVXHXmsOgbATca8P+b665MnPzn50IeSI49MbnWrrlcE0FvT9J4HJXl6kkeUUi5c+fWLc15X5254/wfy4L85JQfnhiTJo59/jDoGwJ5GuT/mhhuSpz0tefe7J9ePOabb9QD03Kone9daP5ykLGAty+PjH8+uxz0+P1J/mCS5vhya4595VMeLAmCZjHJ/3LUr2bYtefObd9927LHdrQdgADSffV10UeqJJ+aQH1xz003X/cRP5aBDvVEZgBGrNXn+85Mzz9z7dgMZwIYYyPZ0ySXJCSekXHHFXjff6kE2GwBG7rTTkle8Yv/bnbIIsCEGsht97WvJIx+ZfOtb+91VjrXZADBiL3/5ZCDb16ZNydFHL349AANiIEuSyy9PTjgh+epXm+93OgYAY3XGGcnzntd830/+ZPIjP7LY9QAMjIHs2muTU05JvvGN1MMOa36M0zEAGKN3vnPyvrHDD08OOWT/+71gCbBhBrLDDkvOOy+56qr8361/t//9P/qjyRFHLH5dANC1xz42ufLKya+mUxO9YAmwYQayFTt/sDNHvOZPb7r+1Yf9yuTCMcckZVyfagwAeznnnOSiiyaXjzoqecADJpcVMoAN81nuKz72nLPy4J1fTpJ84Wb3zlHvOzN57mGTH4AJAGO1a9feH+jx4hcnD31octxxChnADBjIsn8du+I5L8lBBx+U/OVfJh/7WIcrA4CO7VvHtm5NDj44Ofvs5G5363ZtAAPglMVM6thP7FHH7vc/nzC542Y3Sx72sO4WBgBdaqpjB6+8lnvCCclBDiMANmr030lb6xgAjF1THQNgpkY/ebTWMQAYswPVMQBmZtQDmToGAC3UMYCFGPX0oY4BQAN1DGBhRjuQqWMA0EIdA1iY0U4g6hgANFDHABZqlAOZOgYALdQxgIUa5RSijgFAA3UMYOFGN5CpYwDQQh0DWLjRTSLqGAA0UMcAOjGqgUwdA4AW6hhAJ0Y1jahjANBAHQPozGgGMnUMAFqoYwCdGc1Eoo4BQAN1DKBToxjI1DEAaKGOAXRqFFOJOgYADdQxgM4NfiBTxwCghToG0LnBTybqGAA0UMcAlsKgBzJ1DABaqGMAS2HQ04k6BgAN1DGApTHYgUwdA4AW6hjA0hjshKKOAUADdQxgqQxyIFPHAKCFOgawVAY5pahjANBAHQNYOoMbyNQxAGihjgEsncFNKuoYADRQxwCW0qAGMnUMAFqoYwBLaVDTijoGAA3UMYClNZiBTB0DgBbqGMDSGszEoo4BQAN1DGCpDWIgU8cAoIU6BrDUBjG1qGMA0EAdA1h6vR/I1DEAaKGOASy93k8u6hgANFDHAHqh1wOZOgYALdQxgF5YdXoppZxZSrm8lPKZRSxoLdQxALq0tHukOgbQG9PkpNckecyc17Fm6hgAS+A1WcI9Uh0D6I9VJ5ha6weTfHcBa1kTdQyAri3lHqmOAfTKzJJSKWVbKWV7KWX7jh07ZvVlG6ljAPTFIvfHJOoYQM/MbIqptZ5ea91Sa92yefPmWX3ZRuoYAH2xyP1RHQPon95lJXUMAFqoYwC907tJRh0DgAbqGEAvTfOx929I8tEk9yylXFZK+Y35L6uZOgbAMlmmPVIdA+inVV86q7UuzXf0jz3nrDxYHQNgSSzNHqmOAfRWb/KSOgYALdQxgN7qzUTjvWMA0EAdA+i1Xgxk6hgAtFDHAHqtF1ONOgYADdQxgN5b+oFMHQOAFuoYQO8t/WSjjgFAA3UMYBCWeiBTxwCghToGMAhLPd2oYwDQQB0DGIylHcjUMQBooY4BDMbSTjjqGAA0UMcABmUpBzJ1DABaqGMAg7KUU446BgAN1DGAwVm6gUwdA4AW6hjA4CzdpKOOAUADdQxgkJZqIFPHAKCFOgYwSEs17ahjANBAHQMYrKUZyNQxAGihjgEM1tJMPOoYADRQxwAGbSkGMnUMAFqoYwCDthRTjzoGAA3UMYDB63wgU8cAoIU6BjB4nU8+6hgANFDHAEah04FMHQOAFuoYwCh0Ov2oYwDQQB0DGI3OBjJ1DABaqGMAo9HZBKSOAUADdQxgVDoZyNQxAGihjgGMSidTkDoGAA3UMYDRWfhApo4BQAt1DGB0Fj4JqWMA0EAdAxilhQ5k6hgAtFDHAEZpodOQOgYADdQxgNFa2ECmjgFAC3UMYLQWNhGpYwDQQB0DGLWFDGTqGAC0UMcARm0hU5E6BgAN1DGA0Zv7QKaOAUALdQxg9OY+GaljANBAHQMgcx7I1DEAaKGOAZA5D2TqGAA0UMcAWDG3gUwdA4AW6hgAK+Y2IaljANBAHQNgD3MZyGqt6hgANFHHANjDXF6Su/b/fTc/sfMrSdQxANiLOgbAHuayCxz67W/edFkdA4AVV1yRXHrp5LI6BkDmdMrioflhEnUMAPbyzd0vWKpjACRTDmSllMeUUr5QSrmklPL7035xdQyAIVvz/vj9709+V8cAWLHqtFRK2ZTk75KcmOSnk2wtpfz0an9OHQNgyNa7PyZRxwC4yTT56n5JLqm1XlprvT7J2UlOXu0PqWMADNza98dNm5J73EMdA+Am07w8d6ckX9vj+mVJfn7fB5VStiXZtnL1hw942ZM/k5dtfIFL5vZJvt31IuZgqM8rGe5z87z6Z6jP7Z5dL6BD69ofyyWXfCaHHLKA5S3UUP/7Tob73Ib6vJLhPjfPq3+m2iNndr5ErfX0JKcnSSlle611y6y+9rLwvPpnqM/N8+qfoT63Usr2rtew7OyP/TbU5zbU55UM97l5Xv0z7R45zTmFX09yxB7X77xyGwCMmf0RgA2bZiD7ZJKjSil3LaUcmuSpSd6txmufAAAEz0lEQVQ+32UBwNKzPwKwYaueslhr3VlKeXaS9yTZlOTMWutnV/ljp89icUvI8+qfoT43z6t/hvrchvq8VmV/3MtQn1cy3Oc21OeVDPe5eV79M9VzK7XWeS8EAACABj6XHgAAoCMGMgAAgI7MdCArpTymlPKFUsolpZTfn+XX7lIp5cxSyuWllM90vZZZKqUcUUr5QCnlc6WUz5ZSntv1mmahlHKzUsonSimfXnlep3W9plkqpWwqpXyqlPKOrtcyS6WUr5RSLiqlXDikj1Ivpdy6lPKWUsrnSykXl1Ie0PWaZqGUcs+Vf1c3/rqqlHJq1+taZkPcI+2P/WOP7J+h7o/JMPfI9eyPM3sPWSllU5IvJjkhkx+O+ckkW2utn5vJX9ChUspDklyT5J9rrcd0vZ5ZKaXcMckda60XlFIOT3J+kif0/d9ZKaUkOazWek0p5ZAkH07y3Frrxzpe2kyUUp6XZEuSH621ntT1emallPKVJFtqrYP64ZCllNcm+VCt9VUrn8R3i1rrlV2va5ZWvv9/PcnP11q/2vV6ltFQ90j7Y//YI/tnqPtjMvw9ctr9cZaF7H5JLqm1XlprvT7J2UlOnuHX70yt9YNJvtv1Omat1vrNWusFK5evTnJxkjt1u6qNqxPXrFw9ZOXXID69ppRy5ySPTfKqrtfC6kopt0rykCRnJEmt9fohbTR7OD7Jlw1jBzTIPdL+2D/2SJbFSPbIqfbHWQ5kd0rytT2uX5aBfPMag1LKXZIcl+Tj3a5kNlZOWbgwyeVJ3ldrHcTzSvKKJC9MsqvrhcxBTfLeUsr5pZRtXS9mRu6aZEeSV6+cQvOqUsphXS9qDp6a5A1dL2LJ2SN7amj7Y2KP7KEh7o/JOPbIqfZHH+pBSim3TPLWJKfWWq/qej2zUGu9odZ6nyR3TnK/UkrvT6UppZyU5PJa6/ldr2VOHlxrvW+SE5M8a+VUqL47OMl9k/xDrfW4JNcmGcR7h260corJ45O8ueu1wKwNcX9M7JE9NMT9MRn4HrmW/XGWA9nXkxyxx/U7r9zGEls5f/ytSc6qtb6t6/XM2kr6/kCSx3S9lhl4UJLHr5xLfnaSR5RSXtftkman1vr1ld8vT3JOJqd49d1lSS7b49Xnt2Sy+QzJiUkuqLV+q+uFLDl7ZM8MfX9M7JF9MdD9MRn+Hjn1/jjLgeyTSY4qpdx1ZSJ8apK3z/DrM2Mrb+w9I8nFtda/6no9s1JK2VxKufXK5Ztn8ib6z3e7qo2rtb6o1nrnWutdMvn/699rrU/reFkzUUo5bOWN81k5XeFRSXr/qW211v9O8rVSyj1Xbjo+Se8/FGAfW+N0xWnYI3tkqPtjYo/sm6Huj8ko9sip98eDZ/U31lp3llKeneQ9STYlObPW+tlZff0ulVLekORhSW5fSrksyUtqrWd0u6qZeFCSpye5aOVc8iT5g1rr/+lwTbNwxySvXflkm4OSvKnWOpiPvx2oOyQ5Z3IMlIOTvL7W+u5ulzQzz0ly1spB+KVJfq3j9czMysHBCUl+q+u1LLuh7pH2x16yR/bLkPfHZKB75Fr3x5l97D0AAABr40M9AAAAOmIgAwAA6IiBDAAAoCMGMgAAgI4YyAAAADpiIAMAAOiIgQwAAKAj/x9T5uNA6c/NTgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 1080x1080 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, axes = plt.subplots(ncols=2, sharex=True, sharey=False, figsize=(15, 15))\n",
    "axes[0].set(adjustable='box', aspect='equal')\n",
    "axes[1].set(adjustable='box', aspect='equal')\n",
    "\n",
    "axes[0].axis([0, 7, 0, 7])\n",
    "axes[0].quiver([0, 0], [0, 0], [3, 1], [6, 2], angles='xy', scale_units='xy', scale=1, color=['blue', 'red'])\n",
    "#axes[0].plot([3], [6], marker='o', markersize=5, color='black')\n",
    "axes[0].annotate('Vector [3, 6]', xy=(3, 6), xytext=(3.1, 6.1), ha='left', color='blue')\n",
    "\n",
    "axes[1].axis([0, 7, 0, 7])\n",
    "axes[1].quiver([0, 1, 2], [0, 2, 4], [1, 1, 1], [2, 2, 2], angles='xy', scale_units='xy', scale=1, color=['red', 'red', 'red'])\n",
    "#axes[1].plot([3], [6], marker='o', markersize=5, color='black')\n",
    "axes[1].annotate('Vector [3, 6]', xy=(3, 6), xytext=(3.1, 6.1), ha='left', color='red')\n",
    "\n",
    "#f1, a1 = plt.subplot(121, aspect='equal')\n",
    "#plt.quiver([0, 0], [0, 0], [3, 1], [6, 2], angles='xy', scale_units='xy', scale=1, color=['blue', 'red'])\n",
    "\n",
    "#f2, a2 = plt.subplot(122, aspect='equal')\n",
    "#plt.quiver([0, 0], [0, 0], [3, 1], [6, 2], angles='xy', scale_units='xy', scale=1, color=['blue', 'red'])\n",
    "#v1 = mpatches.Patch(color='red', label='Vector [1, 2]')\n",
    "#v2 = mpatches.Patch(color='blue', label='Vector [3, 6]')\n",
    "#plt.legend(handles=[v1, v2])\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Scalar division of a vector\n",
    "\n",
    "You usually won't find references to scalar division in resources, because you technically _can't_ divide a vector. But, for sake of completeness, I'm including it here. Recall from arithmetic that dividing by some scalar $s$ is equivalent to multiplying by $\\frac{1}{s}$. Using this idea, we can represent _scalar division_ as scalar multiplication of $\\frac{1}{s}$:\n",
    "\n",
    "$$\\begin{bmatrix}2 \\\\ 4\\end{bmatrix} \\div 2 = \\frac{1}{2}*\\begin{bmatrix}2 \\\\ 4\\end{bmatrix} = \\begin{bmatrix}1 \\\\ 2\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Multiplication of vectors\n",
    "It would be reasonable to conclude that we can multiply 2 vectors together simply by multiplying their components. However, this is **not** the case. Vector multiplication is a more involved process that we'll talk about later. For now, we'll focus on addition, subtraction, and scalar multiplication."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Generalization\n",
    "\n",
    "So far, our operations have been limited to vectors that only have 2 components, but all the operations in this guide apply to vectors of arbitrary size (i.e., any number of components). For example, here is addition with vectors that have 5 components:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}1 \\\\ 2 \\\\ 0 \\\\ 3 \\\\ 4\\end{bmatrix} + \\begin{bmatrix}3 \\\\ 1 \\\\ 2 \\\\ 0 \\\\ 1\\end{bmatrix} =\n",
    "\\begin{bmatrix}4 \\\\ 3 \\\\ 2 \\\\ 3 \\\\ 5\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Generally, we can add vectors with $n$ components by component-wise addition:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}a_1 \\\\ a_2 \\\\ \\vdots \\\\ a_n\\end{bmatrix} + \\begin{bmatrix}b_1 \\\\ b_2 \\\\ \\vdots \\\\ b_n\\end{bmatrix} = \\begin{bmatrix}a_1 + b_1 \\\\ a_2 + b_2 \\\\ \\vdots \\\\ a_n + b_n\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Similarly, we can multiply a vector with $n$ components by a scalar number, $c$:\n",
    "\n",
    "$$\n",
    "c * \\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\vdots \\\\ v_n\\end{bmatrix} = \\begin{bmatrix}c * v_1 \\\\ c * v_2 \\\\ \\vdots \\\\ c * v_n\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Unfortunately, it quickly becomes impossible to visualize vectors with an increasing number of components. However, the beautiful thing about linear algebra is that the concepts scale to vectors of any size. It's just as easy to add 2 vectors in 10-dimensions as it is 2 vectors in 2-dimensions. This neat aspect is seen all throughout the subject."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vectors with different number of components?\n",
    "\n",
    "You may have noticed above that addition and subtraction were performed on vectors that have the same number of components. It's impossible to perform component-wise operations on vectors with a different number of components. For instance:\n",
    "\n",
    "$$\\begin{bmatrix}2 \\\\ 1\\end{bmatrix} - \\begin{bmatrix}1 \\\\ 3 \\\\ 5\\end{bmatrix} = \\enspace?$$\n",
    "\n",
    "This operation is not allowed because $\\begin{bmatrix}2 \\\\\\ 1\\end{bmatrix}$ and $\\begin{bmatrix}1 \\\\\\ 3 \\\\\\ 5\\end{bmatrix}$ have a different number of components. We'll need to take this into account when implementing these operations."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Implementing vector operations\n",
    "\n",
    "Let's see how we might go about implementing vector addition, subtraction, and scalar multiplication from above. First, let's define some vectors to work with:\n",
    "\n",
    "$$\\vec{a}=\\begin{bmatrix}1 \\\\ 2\\end{bmatrix}, \\enspace \\vec{b}=\\begin{bmatrix}3 \\\\ 4\\end{bmatrix}$$\n",
    "\n",
    "Next, we need some boilerplate and helper methods:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Some boilerplate code\n",
    "from functools import reduce\n",
    "\n",
    "# Convert a vector into a comma-separated string (e.g., [1, 2])\n",
    "def v_to_str(v):\n",
    "    return '[%s]' % (', '.join(map(str, v)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, let's define $\\vec{a}$ and $\\vec{b}$ in code:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Define vectors a and b\n",
    "a = [1, 2]\n",
    "b = [3, 4]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first method we'll implement is a `size` method for getting the size of a vector (i.e., number of components):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Size of [1, 2] = 2\n"
     ]
    }
   ],
   "source": [
    "# Returns the number of components in a vector\n",
    "def size(v):\n",
    "    return len(v)\n",
    "\n",
    "a_size = size(a)  # Should be 2\n",
    "\n",
    "# Note, v_to_str is a helper method to convert a Python list into a string\n",
    "print('Size of %s = %d' % (v_to_str(a), a_size))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, let's work on addition and subtraction:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 2] + [3, 4] = [4, 6]\n",
      "[1, 2] - [3, 4] = [-2, -2]\n"
     ]
    }
   ],
   "source": [
    "# Recall, in order to do component wise operations, the vectors need to be the same size\n",
    "def check_size_compatible(v1, v2):\n",
    "    if size(v1) != size(v2):\n",
    "        raise ValueError('Vectors are different sizes')\n",
    "\n",
    "# Adding 2 vectors component by component\n",
    "def add(v1, v2):\n",
    "    check_size_compatible(v1, v2)\n",
    "    return [v[0] + v[1] for v in zip(v1, v2)]\n",
    "\n",
    "# Subtracting 2 vectors component by component\n",
    "def subtract(v1, v2):\n",
    "    check_size_compatible(v1, v2)\n",
    "    return [v[0] - v[1] for v in zip(v1, v2)]\n",
    "\n",
    "a_plus_b = add(a, b)        # > [4, 6]\n",
    "a_minus_b = subtract(a, b)  # > [-2, -2]\n",
    "\n",
    "print('%s + %s = %s' % (v_to_str(a), v_to_str(b), v_to_str(a_plus_b)))\n",
    "print('%s - %s = %s' % (v_to_str(a), v_to_str(b), v_to_str(a_minus_b)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, scalar multiplication. Recall above that multiplication can be seen as multiplying each component by a scalar number or repeated addition, so I've included both:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3 * [1, 2] = [3, 6]\n",
      "3 * [1, 2] = [3, 6]\n"
     ]
    }
   ],
   "source": [
    "# Multiply a vector and a scalar number\n",
    "def multiply(vector, num):\n",
    "    return [num * comp for comp in vector]\n",
    "\n",
    "# Multiply a vector and a scalar number using repeated addition (using functools.reduce)\n",
    "def multiply_alt(vector, num):\n",
    "    return reduce(add, [vector] * num)\n",
    "\n",
    "scalar = 3\n",
    "a_times_3 = multiply(a, scalar)  # > [3, 6]\n",
    "a_times_3_repeated_addition = multiply_alt(a, scalar)  # > [3, 6]\n",
    "\n",
    "print('%s * %s = %s' % (scalar, v_to_str(a), v_to_str(a_times_3)))\n",
    "print('%s * %s = %s' % (scalar, v_to_str(a), v_to_str(a_times_3_repeated_addition)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To demonstrate that these operations are not limited to vectors of size 2, here are a few examples using vectors with more components:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[1, 2, 3, 4] + [2, 0, 1, 3] = [3, 2, 4, 7]\n",
      "[1, 2, 3, 4] - [2, 0, 1, 3] = [-1, 2, 2, 1]\n",
      "3 * [1, 2, 3, 4] = [3, 6, 9, 12]\n"
     ]
    }
   ],
   "source": [
    "# Operations with vectors of size 4\n",
    "a4 = [1, 2, 3, 4]\n",
    "b4 = [2, 0, 1, 3]\n",
    "\n",
    "a4_plus_b4 = add(a4, b4)       # > [3, 2, 4, 7]\n",
    "a4_minus_b4 = subtract(a4, b4) # > [-1, -2, -2, 1]\n",
    "a4_times_3 = multiply(a4, 3)   # > [3, 6, 9, 12]\n",
    "\n",
    "print('%s + %s = %s' % (v_to_str(a4), v_to_str(b4), v_to_str(a4_plus_b4)))\n",
    "print('%s - %s = %s' % (v_to_str(a4), v_to_str(b4), v_to_str(a4_minus_b4)))\n",
    "print('%s * %s = %s' % ('3', v_to_str(a4), v_to_str(a4_times_3)))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Awesome work! We're starting to really build out a good base of vector utilities."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Negativity\n",
    "\n",
    "One minor thing worth mentioning is that we've been referencing and visualizing vectors that are positive and live in the first quandrant. That is more out of convenience. Vectors with negative components exist and are first-class citizens along with their non-negative brethern. For example, consider the following operation:\n",
    "\n",
    "$$\\begin{bmatrix}2 \\\\ 1\\end{bmatrix} - \\begin{bmatrix}1 \\\\ 3\\end{bmatrix} = \\begin{bmatrix}1 \\\\ -2\\end{bmatrix}$$\n",
    "\n",
    "The result contains a negative component, and can be visualized as:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "hideCode": true,
    "hideOutput": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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55XjyySfjfjBrx6yYj6WU86SU+VLKOVLKEe/J5aNUZfz48dQlGIPbKyhOz0Knbe/KlStt2WesWhN88MEHfW16t27dCgDweDy44IILsH79eixcuDAm+x0OyTmR0Lk1BrS3t1OXYIza2lrqEozh9Cx02vZ+/vnncW3bO5yzzjoLjY2NA7520kkn9f1HtGTJEnR3d2PZsmX43ve+13eKKN5IesVwSwElFg2AEtWsWbOoSzBGXLOYAsDOJVY12vYuW7ZswMLWZ599Nj755BOcfPLJAKz/ADZv3jygbW9SUhLmzZuHp556CsuXL8cPf/hDPPjgg/jjH/+Ip59+GldffTU6Ozvx5S9/GU8++aR2+c3NzViwYAE8Hg/GjBmDTZs2oaKiAhkZGaiurh70NFB/zz//PN555x20tLTgqaeeAgA89dRTgy68ESskAzvflKNwEzCltLQU8+fPpy7DCHHNguiv4De/+c1BZ4Vdf/31uP766/s+D83pvuKKK3DFFVcMeO6iRYtQUVEx4Gvvv3/oTe9vv/122HXl5OSgoaHhkK/v3LkTF1988Yi9ay677DJcdtllYe8vFnihDWJTp06lLsEYPKgrnIUSr7a9Iy3KMWfOHGzcuDGqfX33u9/F9u3bY37Wgo/YifHSeEpxcTH3IQ/iLJQDBw7EdHCPZ9veZ599Ni774SN2Yk6f/aCDBzIl1lkk0o1xsT5iN1G0Px9ezJpYU1MTdQnGCPUzYbHNIj09HS0tLQkzuIfOsY8WUkq0tLREdbqGl8YjlpOTQ12CMeI5a8B0scwitDBzvG9zj5SUEkII6jLiKj09ParWzSQDOzfOV7g9q1JZWYnZs2dTl2GEWGaRkpKC6dOnx2TbsVBRUYHjjz+euoyEQnIqhhvnK5mZmdQlGCORBptY4ywUzkIfycDe09NDsVsjhe6aYzxDqD/OQuEs9JEM7MnJJGeAjMQL9Soj3dE3mnAWCmehj2Rgd3KvaV3824vCs6UUzkLhLPRxY3Rio+1q/3C4T7/CWSichT6SxHgwU5KSkqhLMAa3c1Y4C4Wz0MenYohxC2Ml1KqVcRb9cRb6+OIpMb5ZS+EWxgpnoXAW+nihDWIej4e6BGMM1ip1tOIsFM5CH8nAnpaWRrFbI/FULmXGjBnUJRiDs1A4C30kAzu3FFBcLhd1CcbYtWsXdQnG4CwUzkIfycDON+Uo3ARMKSgooC7BGJyFwlno47a9xPh2aaW4uJi6BGNwFgpnoY8X2iDGC20ovNCGwlkonIU+PmInxkfsCh+ZKZyFwlno4yN2YnzErvCRmcJZKJyFPpKB3ev1UuzWSHv37qUuwRjl5eXUJRiDs1A4C30kA3s0a/k5TXZ2NnUJxpg5cyZ1CcbgLBTOQh/feUps37591CUYo76+nroEY3AWCmehj3vFEMvIyKAuwRhTpkyhLsEYnIXCWegjGdh7e3spdmskvgtXaW9vpy7BGJyFwlnoIxnYuXG+wr+9KHztReEsFM5CH4+wjDHmMFEP7EKIo4QQbwkhKoQQu4QQ14/0Gl5oQ/H7/dQlGINPSymchcJZ6LPjPIAfwE+llCVCiPEAioUQr0spK4Z6AS8Hp/CvmUpmZiZ1CcbgLBTOQl/UR+xSyiYpZUnw8X4AnwA4crjX8FGqwst+KXyzlsJZKJyFPlvPsQshjgEwD8AHg3xvpRCiSAhR1NraCrfbjaamJjQ2NqKtrQ01NTXwer2oqKhAIBBASUkJANUnoqSkBIFAABUVFfB6vaipqUFbWxsaGxvR1NQEt9uNuro6dHR0oLKyEn6/H2VlZQO2EfqzvLwcPp8PVVVV8Hg8qK+vh8vlgsvlQn19PTweD6qqquDz+fruejt4G2VlZfD7/aisrERHRwfq6uoiek8TJkxw3HuK9OfU1tbmuPcU6c8pJyfHce8p0p9TSkqK495TpD+ncAkpZdhPHnZDQmQA2A5gvZTyz8M9Ny8vT3LzfMs111yDRx55hLoMI5SXl2Pu3LnUZRiBs1A4C0UIUSylXDDS82w5YhdCpAD4E4BnRxrUAV5ooz+++ULhf7wKZ6FwFvrsmBUjADwO4BMp5cZwXsNtexVu26twe1aFs1A4C312HLEvAnA5gDOFEKXBj/OHewG37VW4ba/C7VkVzkLhLPTZMSvmH1JKIaXMl1IWBj9eHe41fMSu8BG7wkdmCmehcBb6eKENYnzErvCRmcJZKJyFPl5og1hzczN1CcYITT1jnEV/nIU+XmiD2OTJk6lLMEZeXh51CcbgLBTOQh/JwO7z+Sh2a6TW1lbqEoxRXV1NXYIxOAuFs9BHMrCnpqZS7NZIEyZMoC7BGLm5udQlGIOzUDgLfSQDO/eKUXiGkOJ2u6lLMAZnoXAW+nihDWL824vCywQqnIXCWegjGWHt6k/jBLxMoNLT00NdgjE4C4Wz0MeHzsT4PzmFF2BROAuFs9DHp2KIpaSkUJdgDL5xTeEsFM5CH188JcY3ayk89VPhLBTOQh/JwM5Hqcr48eOpSzAGt1dQOAuFs9BHMrB3d3dT7NZI7e3t1CUYo7a2lroEY3AWCmehj1sKEJs0aRJ1CcaYNWsWdQnG4CwUzkIfycDON+Uo3ARMKS0tpS7BGJyFwlno47a9xKZOnUpdgjHmz59PXYIxOAuFs9DHR+zEeKENhRdUUDgLhbPQx0fsxPiKv8ILKiichcJZ6OMjdmJNTU3UJRijpKSEugRjcBYKZ6GPj9iJ5eTkUJdgjMLCQuoSjMFZKJyFPpKBvauri2K3RuKWpEplZSV1CcbgLBTOQh8vtEEsMzOTugRjTJ8+nboEY3AWCmehj2Rg5zacyv79+6lLMAbPEFI4C4Wz0EcysCcnJ1Ps1khjx46lLsEYWVlZ1CUYg7NQOAt9JAM791dW+LcXhWdLKZyFwlno48boxIQQ1CUYg/v0K5yFwlnoI0mMBzMlKSmJugRjcDtnhbNQOAt9fCqGGLcwVjo6OqhLMAZnoXAW+vjiKTG+WUvhFsYKZ6FwFvp4oQ1iHo+HugRjNDQ0UJdgDM5C4Sz0kQzsaWlpFLs1Ek/lUmbMmEFdgjE4C4Wz0MctBYi5XC7qEoyxa9cu6hKMwVkonIU+WwZ2IcQTQgiXEGJnOM/nm3IUbgKmFBQUUJdgDM5C4Sz02XXE/hSAc8N9Mt9woPDt0govqKBwFgpnoc+W6SlSyneEEMeE+3yeCaLwQhuKYxdUkDLsj9e2CRyWIfC1bzg0iwg49u9FDMXtHLsQYqUQokgIUdTQ0AC3242mpiY0Njaira0NNTU18Hq9qKioQCAQ6GuuH/rfuqSkBIFAABUVFfB6vaipqUFbWxsaGxvR1NQEt9uNuro6dHR0oLKyEn6/H2VlZQO2EfqzvLwcPp8PVVVV8Hg8qK+vh8vlgsvlQn19PTweD6qqquDz+VBeXj7oNsrKyuD3+1FZWYmOjg7U1dVF9J727NnjuPcU6c/p9ddfd9x7Kv7gA+DUU1H8058Cycko/9GP4Js0CVXLl8Nz3HGoP/98uE45Ba5Fi1Bz4TKsu3kPNj7SgN/8Zoe57ynOf/feffddx72nSH9O4RJSyrCfPOyGrCP2l6WUc0Z67oIFC2RRUZEt+01069atw7p166jLYLG0bx9w5pnAcCsBJSfjl2e+gpu2nQ0A+PWvgWuuiVN9LGEIIYqllAtGeh7JrBid/3mcbu/evdQlGCN05OMYBw4AL70E3HgjMNxc7IICtL5Rgrt3WIP6zJnAKac4LIsoOO7vRRyQ3AKanp5OsVsjZWdnU5dgjJkzZ1KXEL2mJuDll4GtW4E33gCGm9qbnAzcdhuwejV+sToF7e3Wl++9F5g1ywFZ2MQRfy/izK7pjn8A8B6A44QQDUKIHwz3fL7zVNm3bx91Ccaor6+nLkGflEB5ObB+PXDSScC0acDKldbgHhrUU1OB884Djj9eva6wECgqAtauRV1jCn71K+vLp5wCfPObCZpFjHAW+uyaFbNCa6fcK6ZPRkYGdQnGmDJlCnUJ4enpAd55xzoq37oVqKs79DmTJgGLFwNLlgDf+AaQkQEsWACkpFhH6bfcYj0G8POfA6FjnQ0bACESKIs44Cz0kYywvb29FLs1Et+Fq7S3t2PChAnUZQyurQ34v/+zBvK//c26IHqwWbOsgXzJEmDhQqB/S+ZAABg71jpKz8/v+3JJCfDss9bjiy+2jtgBw7OIM85CH8nAzo3zFf7tRTHu2stnn6mj8nffBfz+gd8fMwb42tesgfzCC62rnkOREnjzzb6j9NCXbr7ZepycDPziF+rpxmVBiLPQx6MKYyGBALBjhxrMB+tRkpEBnHuuNZiffz4Q7sXvpKSBR/AAXnvNGusB4Ec/Gv7/BcZ0kAzsvNCG4j/4KHAUIzkt1dlpzV7ZutW64DnY9NOjjlKnWE47DbChO2lvL7BqlfU4IwNYu3bg9/kUncJZ6CMZ2Hk5OIV/zVQyMzPjs6PmZjUl8fXXB5+S+NWvqsG8oMC6ommjZ56xJtMAwM9+BkyePPD7ccsiAXAW+kgGdj5KVXjZL2Xv3r2xuUgmJbBzpzrFsmPHoc9JTQXOOssayBcvBnJz7a8jqLPTmhgDAFOnAjfccOhzYpZFAuIs9JEM7KmpqRS7NdLEiROpSzDG0Ucfbd/GwpmSmJ09cEri+PH27X8YDzwANDZaj++8EzjssEOfY2sWCY6z0EcysPM5M6WlpYW6BGPs3r0bc+fOjXwD7e3WVMThpiQed5w6xXLyyYdc0Iy1L74A7rnHepyXB1x55eDPizoLB+Es9JEM7LzQhsI3XygR/eP97DPgr3+1BvN33hl6SuKFF1ofxx1nT7ERuvtuYP9+6/F991nTHAfDA5nCWegjGdh5oQ2FF9pQiouLR+69HQgAH36oTrHsHGTRrkinJMZYdbXVtREATj/dKm0oYWUxSnAW+kgGdl5oQ+GFNpQh//GGpiT+9a/Wx2BTEnNz1SmW00+3ZUqi3W69Vf1Ccf/9w0+04YFM4Sz08RE7MT5iVwYcmYUzJXH+fDWYFxbaPiXRTh98ALzwgvV4+XLghBOGfz4fpSqchT4+YifGR+xBUuKraWnWffVbt1oj4cFSU60FK0JTEo86Kv51RqB/64CUFKsR5Eh4IFM4C30kAzsvtKE0NzdTl0Cnp8fqwRI8X1529tkoePTRgc/JzgYuuMAazM8+O25TEu30179abxMAfvxj4MtfHvk1ZWVlKCgoiG1hCYKz0McLbRCbfPAth04XmpL4178Cr746YEpi3tNPWw+IpyTaye+37iwFgIkTgTVrwntdXl5e7IpKMJyFPpKB3efzUezWSK2trdQlxF5trZrFMtSUxEWLUH3llZi1aBH5lEQ7PfEEUFlpPb711vAn6FRXV2PWrFmxKyyBcBb6+M5TYo68VTqcKYmHHTZwSuKkScjt6LCmKjpERwdw++3W46OOAv7rv8J/bW4MWxokGs5CH/eKIeaYGUKdncDf/24N5ENNSTzyyIFTEg86Jed2ux21otQvf2lN7gGsG5N07stzWhbR4Cz08UIbxBL6t5e9ewdOSRzsovi8edZAvnTpiFMSnfSPt7nZWuYOsJpDfve7eq93UhbR4iz0kQzsUkqK3RopoZYJlBKoqFCnWD74wPpaf1FMSezp6bG5YDp33AEcOGA9vv9+/eu/TsoiWpyFPl5BiZjx/8kdNCURtbWHPicrS3VJjGJKolMWYKmsBH77W+vx2WdbH7qckoUdOAt9fCqGWEq/NTCN0d6uFm4+aEpin2OPtU6vhKYk2rB2q1NuXFu92lohSQir0VcknJKFHTgLfXzxlJgxN2vV1qouidu3Dz4l8ZRT1MXPGExJbG1txeGHH277duPpH/8AXnzRenz55dZlhUg4IQu7cBb6SAZ2I49SiYynupMyEACKitQpltA6bf2FpiReeKE1JfGII2JaUqK3V+jfOiAtDbjrrsi3lehZ2Imz0EcysHd3d1Ps1kjt7e3x21n/KYkvv6zm4vU3wpTEWKqtrcXs2bPjtj+7/elPwPvvW4+vvx6IZuGfRM/CTpyFPm4pQGzSpEmx3YHOlMQlS6zHRF0SE/nuwu5u69w6YF1LDj2OVCJnYTfOQh+37SVmexOwcKYkpqSoKYmc5DYcAAAgAElEQVQXXmhMl8TS0lLMnz+fuoyIPPaYtZAGAPz850BmZnTbS+Qs7MZZ6OO2vcSmTp0a/UZ6eqyrdqHB/LPPDn1OVtbALokGtjJI1H+8Ho81bx0Apk8Hrr02+m0mahaxwFno4yN2YhEvtLFv38ApiYOdqz/2WHWK5ZRTbJmSGEuJuqDC/fcDbrf1+Be/sGfxpkTNIhY4C318xE5M64p/XZ3qxfL22yRTEmMpEf/xNjYCGzdajxcsAL79bXu2m4hZxApnoY+P2Ik1NTUN/c1wpySec47qkhjjKYmxVFJSknC/dq9dq65H33+/9X+rHRIxi1jhLPTZMrALIc4F8ACAJAC/k1LeO9zz+YhdycnJGfgFr3dgl8TBLq5Om6aOys84I65TEmOpMNK7eYiUlwNPPWU9vuAC60dhl0TLIpY4C31RD+xCiCQADwP4BoAGAB8KIbZKKSuGek3XYAsTj1Jut9uakvjKK9Zgvm3b4FMSCwvVYD5/vtELN0eqsrIyoeYr33KL9UvVmDHAvcMeyuhLtCxiibPQZ8cR+4kAqqWUnwGAEGILgKUAhhzYE7pVrZ3efx+Z//ca8Mgjg09JPOMMqx/L4sXR3e2SIKZPn05dQtjefNO6Zg0A//mfwJw59m4/kbKINc5Cnx1nBI8E8Hm/zxuCXxsSt+EMCmRhf9JCQK6wPs/KshqMvPCCNc3itdesuXOjYFAHopghFGeBALBqlfV47Fjgzjvt30eiZBEPnIW+uF08FUKsBLASACZOnIhVq1aht7cXUkqkpKTA6/Vi/PjxaG9vx6RJk9Dc3IypU6diz549mDZtGpqampCTkwO3243MzEzs378fY8eORU9PD4QQSEpKQnd3N8aNGwePx4OsrCy4XC7k5OT0bSP05969e5GdnY19+/YhIyMDXV1dSA5OBfT7/UhPT0dHRwcmTpyIlpYWTJky5ZBtNDc3Y/LkyWhtbcWECRPQ2dmJ1NTU8N/T1Glo2t6EnQ21uC4/H5lHX4X9x6Rj7Lhx6PnXvyDeey/x3lOUP6fPP/8cRx11lPHvaffu/aitHYujjurBrFkCmzbZ/3NKS0vDgQMHjPw5xfvvns/nw2GHHeao9xTpzyns8TbafuBCiJMBrJNSnhP8fDUASCnvGeo1BQUFsqysLKr9OsLrwI1X3YiN9RuBqQCKAIzifkeNjY048shhf9kj5/MBs2ZZM0+POMK62zQW93olQhbxwlkoQohiKeWCkZ5nx6mYDwEcK4SYLoRIBbAcwFYbtut83wBEfvAiaBOAbwIwpIsvhUTo0//ww9agDlgLVcfqBt5EyCJeOAt9UScmpfQD+DGA1wB8AuB5KeWu4V4jHDijI1JJxycBPwh+8iGsk1WGL6oUK6a3c25rsxalBqybeleujN2+TM8injgLfbb8VyilfFVKOVNK+RUp5fqRns9LXSnd3d3WZNFFwS/8HsD/IyyIUEdHB3UJw7rnHmtwDz2O5XhjehbxxFnoI/kdJ9nwniXxNG7cOCANwJ8AhJos/gzAq3Q1UYl5C+Mo/PvfwIMPWo9PPhm46KLY7s/kLOKNs9BHMrDzQhuKx+OxHkwB8BKAcbBOxayAdWJrFGloaKAuYUi33WZdOAWADRtif3+YyVnEG2ehj2RgT7Oj/Z1DZGVlqU/mAXgq+NgDYAmA1riXRGbGjBnUJQzqo4+A3//eerxsGbBo0fDPt4OpWVDgLPSRDOzcUkBxuVwDv/AtAGuDj6sBXApglKz9vWvXsNfcSUhp3YwkJZCUZJ1bjwcTs6DCWegjGdjHjh1LsVsjHdIEDABuB7As+PgNADfFsSBCBQUF1CUcYts24I03rMcrV8avE7KJWVDhLPSRDOzctlcZ9HbpMQCeAZAf/PwBAI/HryYqxcXF1CUM0NurWgccdpg1bz1eTMuCEmehj2Rg57a9ypALbWTAupgamhBwDYB/xKcmKqYtqPD73wMff2w9XrUKmDIlfvs2LQtKnIU+PmInNmyDo2NgTYNMBtAD4CIA9fGoioZJR2Zer7UoNQDk5AA33hjf/ZuUBTXOQh8fsRMbcWm8/4B1AxMAfAGrIfKB2NZExaQjswcfBEKz7O68E8jIiO/+TcqCGmehj2Rg9w62kMQotXfv3pGftBLAdcHHpQCuhCPbDpQPtvQfAbfbWpQaAI4/3uq3Hm+mZGECzkIfycCe7pCl3OyQnZ0d3hP/B0Bo6bU/ArgrRgURmjlzJnUJAKx+MKH7xu67D6C4UdqULEzAWejjO0+J7du3L7wnpgB4AcCXg5/fDuDPsamJSn09/QWEmhrg17+2Hv/Hf1iLV1EwIQtTcBb6uFcMsQydk7fZsBoih15yOQAHtbWfEs9pJ0NYswYILfAVj9YBQzEhC1NwFvpIBvbe3l6K3RpJ+y7cPAD/C0AA6IR1MfUL28si0d7eTrr/HTuA556zHn/728CJJ9LVQp2FSTgLfSQDOzfOVyL67eVCAKHmyP8GcDEAB5zdorz2EmodAFjteEMXT6nwdSiFs9DHI2yiugVWB0gAeBfWUicOnCkTL6+8Amzfbj2+9lrgK1+hrYexaJAM7LzQhuL3R9jhS8BqMxCa4vtbAL+2pyYqVM3h/H51tD5hgroxiRI3ylM4C30kA3tSUhLFbo0U1a+ZYwG8CCDUR+x6AG9GXxMVnVXY7fTkk8Anwd73q1cDJqzrQJWFiTgLfSQDe8RHqQ4U9bJfuQD+AiAVQC+str81UZdFIqybtWx24ACwNtgmOTcXuP76uJcwKIosTMVZ6CMZ2FNTUyl2a6SJEydGv5GFsE7FANbCHEtgLdSRYI4++ui473PjRqC52Xp8112AKR2lKbIwFWehjxfaINbS0mLPhr4H1be9AsB3YR3BJ5Ddu3fHdX979wL33289njsXuPzyuO5+WPHOwmSchT5eaIOYrTdf3Avg3ODjlwHcZt+m42Hu3Llx3d+ddwKhM2H332+tkGSKeGdhMs5CH7ftJTZs215dSQD+ACC0ys89wc8TRDzbs376KfDoo9bjr38dOOecuO06LNyqVuEs9HHbXmIjtu3VlQmr7UBoIsH3AXxo7y5iJZ7tWVevtlZIAqyjdarWAUPhVrUKZ6GPj9iJ2XrEHjITwHOwfrpdAL4JoMn+3dgtXkdm//wn8Je/WI8vuwyYNy8uu9XCR6kKZ6GPj9iJ2X7EHnI2gF8GH++BtTi24des43FkJiVw883W49RUq0WvifgoVeEs9PFCG8SaQ3PtYuF6AKFFIj6AtWCHwW0Hyspi36ryL38B3nvPevzf/w186Usx32VE4pFFouAs9PFCG8QmT54cu40LAI8AODn4+WYAG2O3u2jl5eXFdPs9PcAtt1iPDz8cuPXWmO4uKrHOIpFwFvpIBnafz0exWyO1trbGdgdpsBbkyA1+vgrA32K7y0hVV1fHdPu//S1QVWU9XrPGGtxNFessEglnoY/vPCU2YcKE2O8kB8BLsHrLBAAsB1AZ+93qys3NHflJEfJ4gHXrrMfHHAP8+Mcx25UtYplFouEs9HGvGGJxmyE0H8BTwcceWG0H2uKz63C53e6YbXvDBuCL4IIk69cDaWkx25UtYplFouEs9PFCG8Ti+tvLtwGEWtJWAbgUgEH/x2otE6hhzx7gl8EZQvPnA8uXx2Q3topVFomIs9AX1QgrhPiWEGKXECIghFgQ7uukNHhqRpzFfZnAO2DNaweA1wHcHN/dD6cntNiozW6/HQhNxNqwAUiE44pYZZGIOAt90f4V3wngIgDv2FDLqBT3/+TGwJodE2q/sQnAE/EtYSixWIBl1y7gieD7O+884Mwzbd9FTPBiNApnoS+qgV1K+YmU8lPtnSbCIVOcpKSkxH+nGbAupmYHP78awD/jX8bBYnHj2i23AIGA1TLgvvts33zM8E18CmehL24jrBBipRCiSAhR1NzcDLfbjaamJjQ2NqKtrQ01NTXwer2oqKhAIBBASUkJAHU7cUlJCQKBACoqKuD1elFTU4O2tjY0NjaiqakJbrcbdXV16OjoQGVlJfx+f9+NDaFthP4sLy+Hz+dDVVUVPB4P6uvr4XK54HK5UF9fD4/Hg6qqKvh8PpSXlw+6jbKyMvj9flRWVqKjowN1dXURvafQ9+P+nqYDxc8UA8lA8XXFwEVA2T/teU+R/pwOfm/R/pzefhs48sgypKf7cc89lZg+Pf7vKdKfk8vlivnfvUT591RbW+u49xTpzylcYqRTAUKIN6AWX+tvjZTypeBz3gZwk5SyKJydzp8/X4bewGi3Zs0arF+/nq6A3wC4Jvh4HoB/ACA6QPJ6vba1dA4EgJNOAoqKrMUzdu+2VkhKFHZmkeg4C0UIUSylHPF65ohH7FLKr0sp5wzy8VKkxXV3d0f6Usdpb2+nLeBqqIH9I1gtCIiubdfW1tq2reeftwZ1ALjhhsQa1AF7s0h0nIU+bilAbJIJKyc/AOD04OPnARD9AjFr1ixbtuPzqXYBkyYBq1bZstm4sisLJ+As9EU73XGZEKIBVjeSV4QQr4XzOm7bq8S0CVi4UgC8AGB68PPbYC2QHWelpaW2bOeRR4DQQd7atYAdy8rGm11ZOAFnoS/aWTF/kVLmSinTpJRTpJRhrUPDV7mVqVOnUpdgmQRrgY7QvSCXAyiPbwnz58+Pehvt7dai1ADwla8AP/pR1JskYUcWTsFZ6OOFNojFZKGNSM0B8HtYXSEPwGo78EX8dm/Hggr33guE+qrdc4/Vcz0R8eISCmehjxfaIBazhTYitRRAaPGJOgCXAIjTte5oF1Sorwc2bbIen3QScMklNhRFhBeXUDgLfXzETqypycA161bD6iMDWPcUXx+f3UY7Bfa226wLp4DVOsC0dUx18HRghbPQx0fsxHJyBrtFgJiA1WYgdGrzN7AW7IixwsLCiF9bWgps3mw9XroUOPVUm4oiEk0WTsNZ6CMZ2Lu6DF98M46MbUk6DsCLAKYEP/9vAG/FdpeVlZE3if/Zz6z1TJOSrPPsiS6aLJyGs9DHC20Qy8zMpC5haEfBmvaYCqu97yUAPovd7qZPnz7ykwaxbZv1AQBXXQU4YdpzpFk4EWehj2Rg5zacyv79+6lLGN7JAB4LPm6FNVMmRiVHMkMoEFA3IB12mFolKdEZNVuKGGehj2RgT05OptitkRKiB8YVAG4MPt4F4DJYS+zZLCsrS/s1zz4LhBaxv+kmwMRLFpGIJAun4iz0kQzs3F9ZSZjfXu4DELr9bCuAtfbvQne2lNdrLUoNAFOmWAO7U/DMMYWz0MeN0YmJRJmTlwxgC4CZwc/XBz+3kW6f/l/9Cvj8c+vxHXcATlpBjdcsUDgLfSSJJcxgFgdJSUnUJYQvE9bReqj3yn8CsPGmQJ1FR1pagF/8wnp83HHAD35gXx0mIFmAxVCchT4+FUMs4VoYHwfgOVh/c7pg3alqUx+zjo6OsJ+7fj2wb5/1+L77AKddttHJwuk4C3188ZRYQt6sdQ6ADcHHjQCWAfBFv9lwWxjX1gIPPWQ9/trXgCVLot+3aYxo52wIzkIfycCecEepMeTxeKhLiMwNsGbLAMD7sBbsiHKBjoaGhrCet2YNELrmnOitA4YSbhajAWehj2RgT0tLo9itkRJ2KpeA1WpgYfDzpwD8T3SbnDFjxojPKSoC/vAH6/EllwALFw7//EQVThajBWehj1sKEHO5XNQlRC4d1p2poWXnbgbwf5FvbteuXcN+X0o1pTE5WV08daKRshhNOAt9JAN7QtyUEydGNgHTkQOrp0w6rJuWlgP4NLJNFRQUDPv9V18Ftm+3Hl9zDXDssZHtJxGMlMVowlno47a9xBxxu/RXATwZfLwPVtuBCNboHm5BBb9ftQ4YP95q0etkvLiEwlno47a9xIxbaCNSywEEF5DG7uDnfr1NDLegwtNPAxUV1uNbbgGOOCKCGhMILy6hcBb6+IidmCOO2EPugjWvHQBeA/AzvZcPdWR24IC1KDUATJsG/OQnkRaYOPgoVeEs9PEROzHHHLED1t+mzQDygp9vhDVbJkxDHZlt2gSE/v+76y5gNPz14aNUhbPQRzKwe71eit0aae/evdQl2Gs8rLYDoVmcPwLwXngvLS8vP+RrLpd1ZykA5OUBV1xxyFMcabAsRivOQh/JwJ6enk6xWyNlZ2dTl2C/LwP4I4AkWAthLwMQxj0mM2daHcaqqtTX7rwTCLWsv/9+a4Wk0SCUBeMsIsF3nhLbF2p44jRnAHgw+HgvrHPvoUsr22D1mTlIfX09AOt0ywsvALt3A48+an3vzDOB886LacVGCWXBOItIcK8YYhlO6jV7sGthtRoAgBIA34fVduDnsBqJHWTKFGuB1Q8+AK6+GrjuOmuaI2AdrTuxdcBQQlkwziISJAN7b28vxW6N5Pi7cB8AcFrw8XMALgXwIYCHD31qe3s72tqsI/XWVuCNN6yvX3wxMNoWqm9vj+BGAIfiLPSRDOzcOF9x/G8vqQBeAHBM8PMXgn9+GPzoJz09HUVFh27iT3+yWggsXQqMlrN4fB1K4Sz08QjLYkfCWiP1LwC+NMj3Bzlq/+CDwTd1/vnAc88Bqan2lceYU/FCG8T8fs3bMxOJAFAHYB2A7YN8fwsAt/q0q6sLO3Yc+rTFi4E//xkYTQdujj9Fp4Gz0EcysCfUcnAx5vhfMy+AddR++SDf8wF4Qn06cWLmIUfsS5dap2JGW6fnzMxM6hKMwVnoIxnYHX2UqmlULPt1OIBnYHWBnHzQ9x4BELyW/umne9G/i/FFFwHPPz86T7847sa1KHAW+kgG9tTR+C91CBMnThz5SU6xFNbR+/J+X6sD8Dfr4b//fXTfl7/1LWDLltE5qAPA0UcfPfKTRgnOQl9UA7sQYoMQolII8bEQ4i9CiLB+Z+JzZkpLSwt1CfE1CcAfYM2OCS1lGbyI+sUXuwEAy5cD//u/wGhenH737t3UJRiDs9AX7RH76wDmSCnzYTVrXR3Oi3ihDWXU3nxxCayj94thrbpUDWzZMhff+Q6webM1vXE0mzt3LnUJxuAs9EU1sEspt0kpQyfM34daJG1Y3LZXcVTbXl2TYR25bwF6nwNWrCjGM8/woA5wq9r+OAt9dp5j/z76zpYeSgixUghRJIQo2rdvH9xuN5qamtDY2Ii2tjbU1NTA6/WioqICgUAAJSUlANQPtaSkBIFAABUVFfB6vaipqUFbWxsaGxvR1NQEt9uNuro6dHR0oLKyEn6/H2VlZQO2EfqzvLwcPp8PVVVV8Hg8qK+vh8vlgsvlQn19PTweD6qqquDz+fo6yx28jbKyMvj9flRWVqKjowN1dXURvaepU6c67j1p/Zx6/SibVYbAj4ETTrCafCX8e7Lh5zRnzhzHvadIf045OTmOe0+R/pzCJaSUwz9BiDdgrWx5sDVSypeCz1kDYAGAi+RIGwQwe/ZsWRFaDmeUW7lyJR577DHqMoxQXFzMvbeDOAuFs1CEEMVSygUjPW/EX3qllF8fYUdXAlgM4KxwBnWAF9roz1ELbUSJ//EqnIXCWeiLdlbMuQBWAVgipQz7xDkvtKE0NzdTl2CM0K/FjLPoj7PQF+059odgrZnzuhCiVAjxm3Be5Pi7LTVMnnzwHTujV15e3shPGiU4C4Wz0BftrJgZUsqjpJSFwY+rR34V4PP5otmto7S2tlKXYIzq6mrqEozBWSichT6+85TYhAkTqEswRm5uWLNlRwXOQuEs9HGvGGI8p19xu90jP2mU4CwUzkIfL7RBjH97URy9TKAmzkLhLPSRjLBhzoocFXiZQKWnp4e6BGNwFgpnoY8PnYnxf3IKL8CicBYKZ6GPT8UQSxnNLQwPwjeuKZyFwlno44unxPhmLYWnfiqchcJZ6CMZ2PkoVRk/fjx1Ccbg9goKZ6FwFvpIBvbu7m6K3Rqpvb2dugRj1NbWUpdgDM5C4Sz0kQzs3FJAmTRp0shPGiVmzZpFXYIxOAuFs9BHMrDzTTkKNwFTSktLqUswBmehcBb6SAZ2vsqtTJ06lboEY8yfP5+6BGNwFgpnoY+P2ImN6qXxDsJLoCmchcJZ6OMjdmJ8xV/hBRUUzkLhLPTxETuxpqYm6hKMEVrvkXEW/XEW+viInVhOzmDLyY5OhYWF1CUYg7NQOAt9JAN7V1cXxW6NxC1JlcrKSuoSjMFZKJyFPl5og1hmZiZ1CcaYPn06dQnG4CwUzkIfycDObTiV/fv3U5dgDJ4hpHAWCmehj2RgT05OptitkcaOHUtdgjGysrKoSzAGZ6FwFvpIBnbur6zwby8Kz5ZSOAuFs9DHjdGJCSGoSzAG9+lXOAuFs9BHkhgPZkpSUhJ1Ccbgds4KZ6FwFvr4VAwxbmGsdHR0UJdgDM5C4Sz08cVTYnyzlsItjBXOQuEs9PFCG8Q8Hg91CcZoaGigLsEYnIXCWegjGdjT0tIodmsknsqlzJgxg7oEY3AWCmehj1sKEHO5XNQlGGPXrl3UJRiDs1A4C30kAzvflKNwEzCloKCAugRjcBYKZ6GP2/YS49ulFV5QQeEsFM5CH7ftJcYLbSi8oILCWSichT4+YifGR+wKH5kpnIXCWeiLamAXQtwlhPhYCFEqhNgmhAjr8JOP2BU+Ylf4yEzhLBTOQl+0R+wbpJT5UspCAC8DWBvOi7xeb5S7dY69e/dSl2CM8vJy6hKMwVkonIW+qAZ2KWX/u2sOAyDDeV16eno0u3WU7Oxs6hKMMXPmTOoSjMFZKJyFvqjPsQsh1gshPgfwXQxzxC6EWCmEKBJCFO3ZswdutxtNTU1obGxEW1sbampq4PV6UVFRgUAg0LeAbej8WklJCQKBACoqKuD1elFTU4O2tjY0NjaiqakJbrcbdXV16OjoQGVlJfx+P8rKygZsI/RneXk5fD4fqqqq4PF4UF9fD5fLBZfLhfr6eng8HlRVVcHn8/UdLRy8jbKyMvj9flRWVqKjowN1dXURvad9+/Y57j1F+nN67733HPeeIv05ffbZZ457T5H+nMrLyx33niL9OYVLSDn8QbYQ4g0Ag022XiOlfKnf81YDSJdS3j7STufNmyc/+uijsIt0stWrV+Oee+6hLsMIHo8HEyZMoC7DCJyFwlkoQohiKeWCkZ43YjcuKeXXw9znswBeBTDiwN7b2xvmJp2P78JV2tvb+R9wEGehcBb6op0Vc2y/T5cCCGs5cW6cr3CnS4WvvSichcJZ6It2VLlXCHEcgACAfwO4OvqSGGOMRSOqgV1KeXEkr+OFNhS/309dgjH4tJTCWSichT6ScyK8HJzCv2YqmZmZ1CUYg7NQOAt9JAM7H6UqvOyXwjdrKZyFwlnoIxnYU1NTKXZrpIkTJ1KXYIyjjz6augRjcBYKZ6GPF9og1tLSQl2CMXbv3k1dgjE4C4Wz0McLbRCbMmUKdQnGmDt3LnUJxuAsFM5CH7ftJcZtexVuz6pwFgpnoY8X2iDGbXsVbs+qcBYKZ6GPj9iJ8RG7wkdmCmehcBb6+IidGB+xK3xkpnAWCmehj2Rg54U2lObmZuoSjBFqoco4i/44C30kAzvfbalMnjyZugRj5OXlUZdgDM5C4Sz0kQzsPp+PYrdGam1tpS7BGNXV1dQlGIOzUDgLfXznKTHuM63k5uZSl2AMzkLhLPRxrxhiPENIcbvd1CUYg7NQOAt9JAM7L7Sh8G8vSkZGBnUJxuAsFM5CH8kIO9I6q6MJLxOo9PT0UJdgDM5C4Sz08aEzMf5PTuEFWBTOQuEs9PGpGGIpKSnUJRiDb1xTOAuFs9DHF0+J8c1aCk/9VDgLhbPQRzKw81GqMn78eOoSjMHtFRTOQuEs9JEM7N3d3RS7NVJ7ezt1Ccaora2lLsEYnIXCWejjlgLEJk2aRF2CMWbNmkVdgjE4C4Wz0Mdte4lxEzCltLSUugRjcBYKZ6GP2/YSmzp1KnUJxpg/fz51CcbgLBTOQh8fsRPjhTYUXlBB4SwUzkIfH7ET4yv+Ci+ooHAWCmehj4/YiTU1NVGXYIySkhLqEozBWSichT4+YieWk5NDXYIxCgsLqUswBmehcBb6SAb2rq4uit0aiVuSKpWVldQlGIOzUDgLfbzQBrHMzEzqEowxffp06hKMwVkonIU+koGd23Aq+/fvpy7BGDxDSOEsFM5Cny0DuxDip0IIKYQI6zbK5ORkO3brCGPHjqUuwRhZWVnUJRiDs1A4C31RD+xCiKMAnA2gPtzXcH9lhX97UXi2lMJZKJyFPjuO2P8HwCoAvGJEBIQQ1CUYg/v0K5yFwlnoiyoxIcRSAI1SyjLN10WzW0dJSkqiLsEY3M5Z4SwUzkKfGGlpNiHEGwAGm2y9BsCtAM6WUu4TQtQBWCClHHT+nhBiJYCVwU/nANgZadEOMwkAz3m0cBYKZ6FwFspxUsoRF3EYcWAf8oVCzAXwdwChE2C5APYAOFFKOWzLQiFEkZRyQUQ7dhjOQuEsFM5C4SyUcLOIeHqKlLIcwOR+O6zDMEfsjDHG4oOvSjDGmMPYNqFcSnmMxtMfs2u/DsBZKJyFwlkonIUSVhYRn2NnjDFmJj4VwxhjDkM2sAshviWE2CWECAghRt0VbyHEuUKIT4UQ1UKIW6jroSSEeEII4RJCjOopsEKIo4QQbwkhKoL/Nq6nromKECJdCLFDCFEWzOIO6pqoCSGShBAfCSFeHum5lEfsOwFcBOAdwhpICCGSADwM4DwAswGsEELMpq2K1FMAzqUuwgB+AD+VUs4GsBDAdaP474UPwJlSygIAhQDOFUIsJK6J2vUAPgnniWQDu5TyEynlp1T7J3YigGop5WdSym4AWwAsJa6JjJTyHQCt1HVQk1I2SSlLgo/3w/pHfCRtVTSkpSP4aUrwY9ReEBRC5AK4AMDvwnk+n2OncSSAz/t93oBR+g+YDU4IcQyAeQA+oK2ETvDUQykAF4DXpZSjNgsAm2D15Aqrg2JMB+XojQ0AAAFZSURBVHYhxBtCiJ2DfIzao1PGRiKEyADwJwA/kVJ6qOuhIqXslVIWwrqr/UQhxBzqmigIIRYDcEkpi8N9TUwbo0spvx7L7SewRgBH9fs8N/g1NsoJIVJgDerPSin/TF2PCaSU7UKIt2BdhxmNF9gXAVgihDgfQDqACUKI30spLxvqBXwqhsaHAI4VQkwXQqQCWA5gK3FNjJiw2p4+DuATKeVG6nooCSGOEEJkBh+PBfANAKNy8VMp5WopZW7wJtDlAN4cblAHaKc7LhNCNAA4GcArQojXqGqJNymlH8CPAbwG6wLZ81LKXbRV0RFC/AHAewCOE0I0CCF+QF0TkUUALgdwphCiNPhxPnVRRKYCeEsI8TGsA6HXpZQjTvNjFr7zlDHGHIZPxTDGmMPwwM4YYw7DAztjjDkMD+yMMeYwPLAzxpjD8MDOGGMOwwM7Y4w5DA/sjDHmMP8ftNBO2WZm86UAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(6, 6))\n",
    "axis([-1, 4, -4, 4])\n",
    "grid(linestyle='dotted')\n",
    "quiver([0, 2, 0], [0, 1, 0], [2, -1, 1], [1, -3, -2], angles='xy', scale_units='xy', scale=1, color=['red', 'blue', 'magenta'])\n",
    "v1 = mpatches.Patch(color='red', label='Vector [2, 1]')\n",
    "v2 = mpatches.Patch(color='blue', label='Vector [-1, -3]')\n",
    "v3 = mpatches.Patch(color='magenta', label='Vector [1, -2]')\n",
    "legend(handles=[v1, v2, v3])\n",
    "#plt.grid(True, which='both')\n",
    "axhline(y=0, color='k', linewidth=0.5)\n",
    "axvline(x=0, color='k', linewidth=0.5)\n",
    "#plt.xaxis.set_major_locator(ticker.MaxNLocator(integer=True))\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## More than 2 vectors?\n",
    "\n",
    "It may go without saying, but for the sake of explicitness, vector operations can be combined and applied to any number of vectors. For instance, here is an example of addition of 4 vectors:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}2 \\\\ 1\\end{bmatrix} + \\begin{bmatrix}1 \\\\ 2\\end{bmatrix} + \\begin{bmatrix}1 \\\\ 0\\end{bmatrix} + \\begin{bmatrix}3 \\\\ 2\\end{bmatrix} = \\begin{bmatrix}7 \\\\ 5\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Similar to standard arithmetic, we can combine different operations together. When doing so, the same order of operations applies to vectors:\n",
    "\n",
    "$$\n",
    "2 * \\begin{bmatrix}2 \\\\ 1\\end{bmatrix} + 3 * \\begin{bmatrix}1 \\\\ 2\\end{bmatrix} = \\begin{bmatrix}4 \\\\ 2\\end{bmatrix} + \\begin{bmatrix}3 \\\\ 6\\end{bmatrix} = \\begin{bmatrix}7 \\\\ 8\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "This combination of scalar multiplication and addition is known as a _linear combination_. The name may sound complicated, but as we'll see below, it's a pretty simple concept."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linear combinations\n",
    "\n",
    "As shown, we can multiply a vector $\\vec{v}$ by a scalar number $c$:\n",
    "\n",
    "$$\n",
    "\\vec{v} = \\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\dots \\\\ v_i\\end{bmatrix}, \\enspace\\enspace c\\vec{v} = c\\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\dots \\\\ v_i\\end{bmatrix} = \\begin{bmatrix}cv_1 \\\\ cv_2 \\\\ \\dots \\\\ cv_i\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "And, add (or subtract) two vectors $\\vec{u}$ and $\\vec{v}$:\n",
    "\n",
    "$$\n",
    "\\vec{u} = \\begin{bmatrix}u_1 \\\\ u_2 \\\\ \\dots \\\\ u_i\\end{bmatrix}, \\enspace \\vec{v} = \\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\dots \\\\ v_i\\end{bmatrix} \\hspace 3em\n",
    "\\vec{u} + \\vec{v} = \\begin{bmatrix}u_1 + v_1 \\\\ u_2 + v_2 \\\\ \\dots \\\\ u_i + v_i\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "We can combine these 2 operations into one step called a **linear combination**. Let's create a linear combination with the above vectors $\\vec{u}$ and $\\vec{v}$ and scalar numbers $c$ and $d$:\n",
    "\n",
    "$$\n",
    "\\vec{u} = \\begin{bmatrix}u_1 \\\\ u_2 \\\\ \\dots \\\\ u_i\\end{bmatrix}, \\enspace \\vec{v} = \\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\dots \\\\ v_i\\end{bmatrix} \\hspace 3em\n",
    "c\\vec{u} + d\\vec{v} = \\begin{bmatrix}cu_1 + dv_1 \\\\ cu_2 + dv_2 \\\\ \\dots \\\\ cu_i + dv_i\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "A linear combination is just the joint step of multiplying the vectors by a scalar number and then adding them together. Let's take an example by finding the linear combination of the two vectors $\\vec{a}$ and $\\vec{b}$ and scalars $c$ and $d$:\n",
    "\n",
    "$$\n",
    "\\vec{a} = \\begin{bmatrix}1 \\\\ 2\\end{bmatrix}, \\enspace \\vec{b} = \\begin{bmatrix}3 \\\\ 2\\end{bmatrix}, \\enspace c = 2, \\enspace d = 1 \\hspace 3em c\\vec{a} + d\\vec{b} = 2\\begin{bmatrix}1 \\\\ 2\\end{bmatrix} + 1\\begin{bmatrix}3 \\\\ 2\\end{bmatrix} = \\begin{bmatrix}5 \\\\ 6\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Changing the values of $c$ and $d$ will give us another linear combination:\n",
    "\n",
    "$$\n",
    "c = 1, \\enspace d = 3 \\hspace 3em c\\vec{a} + d\\vec{b} = 1\\begin{bmatrix}1 \\\\ 2\\end{bmatrix} + 3\\begin{bmatrix}3 \\\\ 2\\end{bmatrix} = \\begin{bmatrix}10 \\\\ 8\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "If we allow $c$ and $d$ to be an arbitrary real number, we can express the linear combination of $\\vec{a}$ and $\\vec{b}$ as:\n",
    "\n",
    "$$\n",
    "c\\vec{a} + d\\vec{b} = c\\begin{bmatrix}1 \\\\ 2\\end{bmatrix} + d\\begin{bmatrix}3 \\\\ 2\\end{bmatrix} = \\begin{bmatrix}c \\\\ 2c\\end{bmatrix} + \\begin{bmatrix}3d \\\\ 2d\\end{bmatrix} = \\begin{bmatrix}c + 3d \\\\ 2c + 2d\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "Choosing values for $c$ and $d$ will produce some vector that is a combination of $\\vec{a}$ and $\\vec{b}$.\n",
    "\n",
    "Naturally, the concept of a linear combination scales to any number of vectors and scalars:\n",
    "\n",
    "$$\n",
    "(c_1 * \\begin{bmatrix}u_1 \\\\ u_2 \\\\ \\dots \\\\ u_i\\end{bmatrix}) + (c_2 * \\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\dots \\\\ u_i\\end{bmatrix}) + \\dots + (c_j * \\begin{bmatrix}w_1 \\\\ w_2 \\\\ \\dots \\\\ u_i\\end{bmatrix})\n",
    "$$\n",
    "\n",
    "Where $c_1, c_2, \\dots, c_j$ are scalar numbers. Linear combinations are super useful and used throughout linear algebra and this guide, so you should be comfortable with them. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Lengths of vectors\n",
    "\n",
    "Above, I mentioned that vectors have a _magnitude_ (i.e., length), but we don't know how to find it yet. It's often useful to find \"how long\" a vector is and we do this by using the components of the vector. Fortunately, finding the length of a vector is as easy as using the Pythagorean theorem. This becomes evident when we visualize the following 2D vector, $\\vec{v}$:\n",
    "\n",
    "$$\\vec{v} = \\begin{bmatrix}1 \\\\ 3\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "hideCode": true
   },
   "outputs": [
    {
     "data": {
      "image/png": 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mvXtZvHgx0tLSzmvYr732Wtx0000YP348EhISsGDBApw9exbx8fF45JFHMHnyZCQlJeH+++8HACxatAjPPPMMLrvsMmRlZeGf//wnHnzwQSQmJiI1NdV5C2NzWrpbKj8/H7GxsXjuuefw5JNPIjY2FuXl5bDb7Th69GiTXwhSSixZsgQJCQlISEhAXl5eq+dtyaOPPor6+nokJiYiPj4ejz76qPNrY8eOxQ033IDExETccMMNGD16dJPn9+rVC6tWrcLixYuRmJiI8ePHIzMzE2fPnsXs2bORmJiIiRMn4rnnngMAvPDCC9izZw8SExNxxRVX4K9//Wur9fXo0QMTJkzAyJEj8eCDD7brNfXv3x8PP/wwxo4diwkTJmDgwIHO4RqLlBK33XYbnn32WVx00UV4/fXXcfvtt6OmpqZd52jJihUrEBsbi6qqKsTGxuKJJ54AAGRlZbW49+/ll1+OTp06eaRX7amfC7/QnoF4T38kJiZ6/KKCv7r//vupSzDGqVOnpJTtv0ianp4u77vvPu3ztHTx1FfPbw8rC284e/aslFLK+vp6OXv2bPnuu+967Vzt8dOf/lQWFBQ0+7Xc3Fw5aNAg2dDQ4OOqzIR2Xjwl6bEH7C1GLrAu1DG1nHPnzp3x2muvtdmbGjlypLO3217PP/88nn766RZ7iKbw5tLWTzzxBJKTkzFy5EjExcVh3rx5XjtXe/z73/9Gr169mvz7m2++iSuuuAIPP/wwb86jSUg3t2YTQkQA+AJAOBwXY9dKKR9v7TmJiYly//79bp03UCxfvhwrV66kLsMIJ06cwMCBA6nLMAJnoXAWihBir5Sy6bjfBTxxFbMWwNVSygohRCiAr4QQm6WU37R4Ur546sSTtRRewljhLBTOQp/b72/ODf1YCy6Envto9W0Ab7ShlJeXU5dgjJycHOoSjMFZKJyFPo8MXAkhgoUQqQAKAGyRUja9N6qRxvfXdnSu3OIXiOx2OwYPHkxdhjE4C4Wz0OeRhl1K2SClTAYQC2CsEGLkhY8RQtwphNgjhNiTm5uLwsJC5OXlITc3FyUlJcjKykJ1dTUyMjJgt9udC/9YkxL27dsHu92OjIwMVFdXIysrCyUlJcjNzUVeXh4KCwtx4sQJVFRUIDMzEzabzXl/sHUM68/09HTU1tbiyJEjKC8vR3Z2NgoKClBQUIDs7GyUl5fjyJEjqK2tRXp6erPHSEtLg81mQ2ZmJioqKnDixAmXXlNBQUHAvSZXvk9z5uzC7bf/EFCvyZ3v0/79+wPuNbn6fdq1a1fAvSZXv0/t5fbF0yYHFOIxAFVSyhbXVx09erTcs2ePR8/rr5544gnn/bwd1fvvv4/HH3/co6v4MRaI2nvx1O0euxCilxAi+tznkQCmA8hs7Tm8bK/S0TfayMvLw6233orIyEjeUKERzkLhLPR54vaUfgD+KYQIhuMXxWop5abWnsB3gigmbLRBxW63Y8mSJSgsLISUH+A3v7kcmzdTV2UGT24u4e84C32euCtmv5TyMillopRypJSyzV0MuMeudOQe+8qVK7Fly5Zzf4tEQYEZC6KZgHupCmehjzfaINZRe+ypqannLfwUFBSMLl26EFZkFu6lKpyFPpKGXefqbqBrz7ragaaqqgqLFy8+bz5DcHAQKisDc/9JV1h3WjDOwhUkDXtERATFaY1kbdbQkdx///3IzDz/+npQUDAiI/mdnGXo0KHUJRiDs9BH0rDzzFOlrKyMugSf2rBhA1599dUm/z5kyCFceWURQUVmsnZqYpyFK3ijDWKdO3emLsFnTp06hdtvv73Zr02evBsPP8wzki19+vShLsEYnIU+koa9oaGB4rRGcnfzAn+SkZGBlStXYs2aNU2+FhUVhdLSUoKqzMRZKJyFPpKGnddWVjrSu5dp06bh5ptvPm9Rp9dffx29evXC3//+Uyxe3JewOrPwdSiFs9DHLSzzubfffhsAMHDgQPz85z/Hp59+ivBw/s/LmKeQdBd5z1NFd+d4f3f8+HHnxsg33ngjhBBISEhAz55AbW3HGZZqS0caomsLZ6GPpMfO28EpHe1t5urVq52fL1q06LyvdaRhqbZER0dTl2AMzkIfScPe0Xqpramo6FiTcqxhmKFDhyIpKem8r/FtsEpHnLjWEs5CH0kXKSwsjOK0RurWrRt1CT5z6NAhpKamAnD01htvar5wIWCzeW8DZ38zYMAA6hKMwVnoI+mx85iZUlTUcSblvPPOO87Pb7zxxvO+dvfdwNSpra723KEcPnyYugRjcBb6SBr2yMhIitMaqaNMvpBSOodhEhISMGLEiPO+XlUFXHJJAkVpRkpI4CwsnIU+koadl+1VOsqyvd9//z0OHjwIoOlFUwCYOROYNImX7bXwUrUKZ6GPl+0l1lGW7bV660DTYRgLL9ur8FK1Cmehj3vsxDpCj11K6RxfHz16NC655JJmH3f2LPfYLdxLVTgLfdxjJ9YReux79+5FVlYWgJZ76wD32BvjXqrCWejjjTaI5efnU5fgdY2HYRYuXNji4zraPf2tSUtLoy7BGJyFPpL72DvabMvW9O7dm7oEr7Lb7c7ZphMmTGjxnuSlS4GGBr5byhIfH09dgjE4C30kPfba2lqK0xqpuLiYugSv2rlzJ06ePAmg9WGYpUuBCROO+Kgq8x09epS6BGNwFvpIGnaeeap07dqVugSvsi6aBgUFISUlpcXHFRYCERGxvirLeLGxnIWFs9DHa8UQC+Q7hBoaGpybakyePBl9+7a83vqCBcCiRbwImKWwsJC6BGNwFvp4ow1igfzu5YsvvnBeHG5uUtKFeNVPpSNtmdgWzkIfSQsrpaQ4rZECeZtA626Y4OBgzJ8/v83HS8nr9Fvq6+upSzAGZ6GPu87EAvWXXH19PdauXQsAmD59Onr27NnmcwI0CpfwZjQKZ6GPh2KIhYYG5lK1n376qfOOn/YMwwBAcDD/XFh4Ep/CWejji6fEAnWyljUMExYWhnnz5rX5+GXLgIULO84Sxm0J9NtgdXAW+khuQwjUXqorAnEafW1tLdavXw8AmDFjRrs2E7nxRqC6uuNsOtKWjrDURHtxFvpIeuy8BZpSWlpKXYLHffTRRygvLwfQ+qSkxk6eBHbsOOnNsvzK8ePHqUswBmehj5cUINaei4r+xhqGiYyMxJw5c9r1nFtuAYAhuOYa79XlT4YPH05dgjE4C328bC+xQFsErLKyEu+99x4AYM6cOVr3IPMiYIq1NyzjLFzBy/YS69evH3UJHvXBBx84f3G3dxjG0rlz4F1vcNWoUaOoSzAGZ6GPe+zEAm2jDWttmC5dumDGjBlaz+WNNhTeXELhLPRxj51YIF3xLy8vxwcffAAAmDt3rvam5YF4h5CreHMJhbPQxz12Ynl5edQleMzGjRudSzK3d1KS5YEHgHnzsrxRll/at28fdQnG4Cz0kdwVwz12pbUVD/2NNQzTvXt3TJ8+Xeu5c+YAdnucN8ryS8nJydQlGIOz0EfSY6+pqaE4rZECZUnS4uJifPzxxwCA+fPna69aeegQsHnzMW+U5pcyMzOpSzAGZ6GPpMceyEvV6oqOjqYuwSPWr1/vXCpCdxgGAO66y9FjnzXL05X5p7g4fvdi4Sz0kfTYeRlOJVDuBLEmJfXq1QtTpkxx6Ri1tTwj2RJod0u5g7PQR9Kwh4TwTjkW3TtHTHT69Gls27YNAJCSkuLy9zc0lH8uLDExMdQlGIOz0EfSsPP6ykogvHtZt26d83vqyjCMpaGBfy4sfOeYwlno4wWwiQkhqEtwmzUM079/f0yYMMHl4wRAFB7DexYonIU+kve+gdCYeYq/7/OZk5ODr776CgCwcOFCl/8T/uY3QFlZDYBwD1bnv3hpa4Wz0MdDMcT8fQnjNWvWOLf3010bprFp04DLLy/xVFl+jxdEUzgLfXzxlJi/T9ayhmEGDhyIsWPHunyc1FTg1KnenirL7wXics6u4iz08UYbxKwNKfzR8ePHsWvXLgCOi6buDLEtX+74YA45OTnUJRiDs9BH0rCHh/M4qsWfb+WylhAA3BuGsQTCrZ+eMnjwYOoSjMFZ6OMlBYgVFBRQl+Ayq2EfNmwYkpKS3D5eZWWl28cIFAcOHKAuwRichT6Shp17Zoq/LgJ26NAh5842N954o0fudNLZbSnQeeIXZaDgLPTxsr3E/HW6tKeHYYDAWV7BE3hzCYWz0MfL9hLzx402pJR46623AACJiYkYMWKE28d86ikA4I02LLy5hMJZ6OMeOzF/7LGnp6c7l1L1VG/9yiuB8HDumVm4l6pwFvq4x07MH3vs3hiG2bEDALhnZuFeqsJZ6CPpsVdXV1Oc1kinT5+mLkGLlNI5KWn06NG45JJLPHLchx8G7r2XZxha0tPTqUswBmehj6Rhj4iIoDitkXr06EFdgpa9e/fi2DHHTkfurOTYnMhIfidnGTp0KHUJxuAs9PHMU2JlZWXUJWixeuuAY9EvT6qt5fkNluzsbOoSjMFZ6HO7YRdCXCyE+EwIkSGEOCCEuLet5/BaMYo/3bttt9ud4+sTJkzAxRdf7NHj85aJSp8+fahLMAZnoc8TPXYbgAeklCMAjANwjxCi1fvfGhoaPHDawOBPs3B37tzpXLfD08MwAJx7pjKgtLSUugRjcBb63O46SynzAOSd+/ysEOIggP4AMlp6Di+cr/jTuxdrGCYoKAgLFizw6LFXrgSKi6sA8PUXgK9DNcZZ6PNoCyuEGAjgMgDfNvO1O4UQe4QQewoKClBYWIi8vDzk5uaipKQEWVlZqK6uRkZGBux2O/bt2wdA3cO6b98+2O12ZGRkoLq6GllZWSgpKUFubi7y8vJQWFiIEydOoKKiApmZmbDZbEhLSzvvGNaf6enpqK2txZEjR1BeXo7s7GwUFBSgoKAA2dnZKC8vx5EjR1BbW+u8In/hMdLS0mCz2ZCZmYmKigqcOHHCpdckpfSL13Ty5Ens3r0bs2bNwuzZs1FcXOzR71NyMhAeftDY75Ovf/bq6+sD7jW5+n0qLCwMuNfk6vep3aSUHvkA0BnAXgDz23psQkKCZA7Lly+nLqFdPv30UwlAApCvvvqqx4+/ZYuU//lPvseP669++OEH6hKMwVkoAPbIdrTHHhkHEEKEAlgH4D9Synfbery/bwfnSf7yNtMahgkJCcH8+fM9fvwnnwQaGnrgpps8fmi/FB0dTV2CMTgLfZ64K0YAeB3AQSnlc+15Dl8kU/xh26/6+nqsW7cOADB9+nSv7WjDt8Eq/jZxzZs4C32eGGOfAOAWAFcLIVLPfcxs7Ql8W5vSrVs36hLatHXrVhQXFwPw3BICzQkP9493L74wYMAA6hKMwVnoc7thl1J+JaUUUspEKWXyuY8PW3uOP93i521FRUXUJbTJunc9LCwM8+bN89p5qqt5cTjL4cOHqUswBmehjzfaIGb65IuamhqsX78eADBjxgyvvsPo1Ml/Jmt5W0JCAnUJxuAs9PGyvcRMX7b3o48+cm647Y1JSZZXXwV++cvvvXZ8f8NL1SqchT5etpeY6cv2WsMwUVFRmDNnjtfOM2wYMGzYSK8d39/wUrUKZ6GPe+zETO6xV1ZW4r333gMAzJ49G506dfLaud5/H3j++aNeO76/4V6qwlnoI2nYuceumNxj/+CDD5y/hL05DAMAf/oTsHHjYK+ew59wL1XhLPTxRhvE8vPzqUtokTUpqUuXLpgxY4bXz+cP9/T7ijXVnXEWruCNNoj17t2buoRmlZeX48MPHXetzps3zyffM28O9fib+Ph46hKMwVnoI2nYa2trKU5rJGvij2k2btzo/D55exjGwu/klKNH+XqDhbPQR9Kw88xTpWvXrtQlNMsahunevTumTZvmk3OGh4f75Dz+IDY2lroEY3AW+khud+S1YhQT7xAqLi7GJ598AgC44YYbfPKL+F//Ak6ePAXAs7sy+avCwkK/2l3LmzgLfSQNO2+0oZj47uXdd991/vL15towjV18Mc9IbowbMoWz0EfSwjqWFWaAmdsEWpOSevfujSlTpvjonMDatbycs6W+vp66BGNwFvq460zMtF9yp0+fxrZt2wAAKSkpPtu67y9/Af75T57fYLHb7dSTSFh7AAAf00lEQVQlGIOz0EfSsPNQjBIaGkpdwnnWrl3r/I/kq2EYS3Aw/1xYeBKfwlnoI/mfxBdPFdNu8bOGYfr3748JEyb49Nz19fxzYTH1NlgKnIU+kobdtF4qpS5dulCX4JSTk4Mvv/wSALBw4UKfv7MKDzfvQjIVk5ea8DXOQh9Jw85boCmlpaXUJTitWbPG+bmvJiU1xhuwKMePH6cuwRichT6S2x15SQHFW/uHusKalBQXF4cxY8b49Nxr1wJ2O9/uaBk+fDh1CcbgLPTxsr3ETFkE7NixY9i1axcAx0VTxx7lvtOzJ5CTk+rTc5osNZWzsHAW+njZXmL9+vWjLgEAsHr1aufnHhuGOXECGNm+zTNWrQL27x/lmfMGgFGjOAsLZ6GPe+zETNlowxqGGTZsGBITE31+/lWrgBdfPOvz85qKN5dQOAt93GMn5vYV/zffBBITgaQk4JZbXDpEZmamc83rRYsWeXYYxmYDfvpT4NJLgQULgFZ+qZt0hxA13lxC4Sz0cY+dWF5enutPPnAAePJJYNs2IC0N+POfXTqMde864IVJSYcOAXffDRw8CHTtCrzySosPrajgHrtl37591CUYg7PQxz12Yn379nX9ydu2ASkpjiuPABATo30IKaVzGCYxMRGXXnqp6/U05+KLAWui0803A1991eJDebEnJTk5mboEY3AW+kgadr5fWSksLCQ9f3p6OjIzMwF46d71C4d1Whnm4XdyivU9YZyFK3ijDWLR0dGuP/nqq4E1a4CiIsffXZh6bfXWAS+tDZOdDezc6fj8v/8FJk5s9mEffghs3sxrxVji4uKoSzAGZ6GP5H8SL8OpnD3rxrhyfDzwyCPA5MmOi6f336/1dCmlc3x9zJgxGDRokOu1tGTYMODllx0XT0tKgGXLmn1YVBRQWmrGHUImMOVuKRNwFvpIZp76ailYf+D25hJLljg+XLBnzx4cO3YMgJd66wMHAu18G/3KK0BVVT/86leeL8MfxbhwvSRQcRb6SHrsvL6yQvnupfEwzMKFC8nqAIDVq4F163ijDQtfb1A4C308qEnM11P3LXa73TnbdOLEibj4Yvq9RomiMBLvWaBwFvpIEqNqzEwUHEzTS92xYwdycnIA+H5DjZYIwf+BLby0tcJZ6OOhGGJUSxhbF02DgoKwYMECkhouZOL+r1QqKiqoSzAGZ6GPpGHni6cKxWQtm83mHIaZMmWKe5OkPIh7ZopJyzlT4yz08UYbxMrLy31+zs8//xwFBQUAaDbUaM727cDf/naEugxjWMNkjLNwBUnDHh4eTnFaI1HcymUNw4SEhGD+/Pk+P39LBg8eTF2CMTgLhbPQx0sKELN6zr5SX1+PdevWAQCmT5+OHj16+PT8LXn2WWDFCt9mYbIDBw5Ql2AMzkIfyWC325NyAoivx7e3bt3q3PXdlGEYANi0CQB402JLUlISdQnG4Cz08bK9xHw9XdqalBQWFoa5c+f69NxtcWt5hQDDm0sonIU+XraXmNsbbWioqanBhg0bAAAzZ85Et27dfHbu9uCNNhTeXELhLPRxj52YL3vsH330kfMuHJOGYSzcY1e4l6pwFvpIxti5x674ssduDcNERUVh9uzZPjtve0RGApGR3GO3cC9V4Sz0kfTYq6urKU5rpNOnT/vkPJWVlXj//fcBAHPmzEGnTp18ct722rwZ+OMf06nLMEZ6Omdh4Sz0kTTsERERFKc1kq9uN9y0aZNzCMyUtWEuNHToUOoSjMFZKJyFPp55SqysrMwn57EmJXXp0gUzZszwyTl1/P73wP/+L68JYsnOzqYuwRichT7eaIOYLzZwLisrw4cffggAmDdvnpHvmD79FGhoMOsuHUp9+vShLsEYnIU+kh47r+Kn+GIW7saNG1FbWwvAzLthLDabjboEY5SWllKXYAzOQh9Jw84L5yu+ePdiDcPExMRg2rRpXj+fq/jnQjHxXRUVzkIf/08KcEVFRfjkk08AAPPnz0dYWBhxRYwxbyMZ7OaNNhRvDz+sX7/eeQ6Th2F69ACqqmwA+BcPwAvlNcZZ6CNp2Km2gzORt99mWpOSevfujcmTJ3v1XO5Ytw4oL+cxdkt0dDR1CcbgLPSRDMXwRTLFm9t+nT59Gp999hkAICUlxfi7kXw1WcsfcBYKZ6GPpGHncV7FmwtxrV271jnsZfIwDAA89BDw2mtx1GUYY8CAAdQlGIOz0McbbRArKiry2rGtYZj+/fvjyiuv9Np5PGHnTmD7dt/+XPz1r0BCApCcDEycCGRk+PT0rTp8+DB1CcbgLPSRNOy80YbirckXOTk5+OqrrwA4lhDwh1sJO3Xy/mStxm66CUhPB1JTgRUrgPvv9+npW5WQkEBdgjE4C328bC8xby3bu3r1aufnpq4Nc6HWlu197DFg5Ur190ceAf78Z/fO17Wr+ryyEhDCveN5Ei9Vq3AW+njZXmLeWrbXGoaJi4vDmDFjvHIOT2tto41bbwXmzweWLwfsduDtt4Fdu5o+btIkoLnfD88+CzQ3N+vll4HnngPq6oBt29wo3sN4qVqFs9DnkYZdCPEPALMBFEgpR7b1eO6xK97osR87dgy7d+8G4LhoKkzqirYgNhYoLi4C0PxqlwMHOu51/+474PRp4LLLHH+/0Jdf6p33nnscH//9L/Dkk8A//6ldulfs3buXG7RzOAt9nuqxrwLwEoA32/Ng7rEr3uixW0sIAP4zDPPvfwMtNeqW228HVq0C8vMdPfjm6PbYLYsWAcuWtbda7+OGTOEs9HmkYZdSfiGEGNjex/NGG0p+fr7Hj2k17MOHD0diYqLHj+8taWlpre5If/31jrH2+npHD7s5Oj32I0eAIUMcn3/wgfrcBG1l0ZFwFvp8dvFUCHGnEGKPEGJPaWkpCgsLkZeXh9zcXJSUlCArKwvV1dXIyMiA3W7Hvn37AKgLJ/v27YPdbkdGRgaqq6uRlZWFkpIS5ObmIi8vD4WFhThx4gQqKiqQmZkJm82GtLS0845h/Zmeno7a2locOXIE5eXlyM7ORkFBAQoKCpCdnY3y8nIcOXIEtbW1zt1bLjxGWloabDYbMjMzUVFRgRMnTrj0mnr16uXR1/T111+jtLQUU6dOxe23344zZ874/DW58n1avhz4wx/6tPp9ys/Pxrhx1Zg1qxK5ue6/pj/8oQLDhzdgxIhaPPtsAx57LMuYn73Bgwcb+X2i+P/UvXv3gHtNrn6f2k1K6ZEPAAMBfN+ex8bHx0vmcM8993j0eI8//rgEIAHIjIwMjx7bmyZPlnLMmMpWH9PQIGVSkpSHD/umJkoHDx6kLsEYnIUCYI9sRxvLM0+JdW18z52bpJTOYZikpCRceumlHju2L4SHh7f4tYwMYPBg4JprzBoy8ZbY2FjqEozBWegjud2R14pRPHmH0P79+5GZmQnAfy6aNlZfXw+g+QXiRowAjh3zbT2UCgsLfbK7lj/gLPR5pMcuhHgLwE4Aw4QQOUKI21o9qR/MgvQVT7578ce7YRrjVT8VbsgUzkKfp+6KWaz5eE+cNiB4aptAKaVzUtKYMWMwaNAgjxzXV4YOBaqq6gGEUpdiBMe7FwZwFq4wex3XDsBTv+T27NmD48ePAzB/JcfmvPYakJtbAoDnOAC8GU1jnIU+3vOUWGioZ3qoVm8dABYuXOiRY/oaT1xTOAuFs9DHG20Q88RkLbvd7hxfnzhxol/eRXDnncCyZTzGbikuLqYuwRichT6SoRhP9VIDQWsLX7XXjh07kJubC8A/h2EA4PBhwG7vRF2GMby1OJw/4iz0kfTY6+rqKE5rpNLSUrePYQ3DBAUFYcGCBW4fjwpvwKJY10sYZ+EKkobd2xs4+5OePXu69XybzYY1a9YAAKZOneq1jTt8gcdSleHDh1OXYAzOQh9vtEHM3UXAPv/8cxQUFADwz3vXG/Pmxt7+JjU1lboEY3AW+nijDWL9+vVz6/nWMExISAjmz5/viZJIJCcDgPvXGwLFqFGjqEswBmehj3vsxNzZaKOurg7vvvsuAODaa69Fj+Z2nvATK1cCt9zCW6BZeDs4hbPQR9Kwc49dceeK/9atW523gvn7MAzAGyo0xlkonIU+7rETy8vLc/m51r3r4eHhmDt3rqdKInHzzcDMmXy/ssVal5txFq7gMXZiffv2del5NTU1WL9+PQBgxowZ6NatmyfL8rmcHADoTl2GMZIdFx0YOAtXkPTY+X5lpbCw0KXnbd68GWfPbe7pr5OSLsTv5BRr+WXGWbiCN9ogFh0d7dLzrGGYqKgozJ4925MlkeH5DUpcXBx1CcbgLPSRNOy8DKdi9bp1VFZW4v333wcAzJkzB506BcZU/NpanpFsceduqUDDWegjGWMPCeHVgi2RkZHaz9m0aZNz2CJQhmHGjwd4hE6JiYmhLsEYnIU+khaW11dWXHn3Yk1K6tq1K6677jpPl0Ti6aeB3NxiAP2pSzFCVVUVunfni8kAZ+EKXhidmBBC6/FlZWX48MMPAQDz5s0LqHFpXqdf4SwUzkIfSY9dtzELZLr7fG7cuNG5OmagDMMAwA03ALW1Mdi0iboSM/DS1gpnoY/kVyEPxSi6SxhbwzAxMTGYNm2aN0oiUVQEFBbyXrgWXhBN4Sz0kTTsfPFU0ZmsVVRUhC1btgAAbrjhhoDryQTa63GHu8s5BxLOQh9vtEGsvLy83Y999913ndsKBsLaMBeqra2lLsEYOY6puAychStIGvbw8HCK0xpJ51YuaximT58+mDJlipcqouPKrZ+BavDgwdQlGIOz0MdLChCzNsloS35+PrZv3w4ASElJ0b7oarprrgFGjnRv05FAcuDAAeoSjMFZ6CMZ7OaemdLeRcDWrl3rvOgciMMwjz4KAO5tOhJIkpKSqEswBmehj5ftJdbe6dLW2jCxsbG48sorvVkSGd5QQeEsFM5CHy/bS6w9G22cPHkSX331FQBg4cKFATlhY8YMALgcmzdTV2IG3lxC4Sz0cY+dWHt67KtXr3Z+HkiTkhqrrgYKCvQXRAtU3EtVOAt9vDUesfb02K1hmEGDBmH06NHeLolMly68mbWFe6kKZ6GPpGGvrq6mOK2RTp8+3erXs7KysHv3bgCOi6aBvBxDZSXPMLSkp6dTl2AMzkIfScMeSAtXuatHjx6tfr0jDMNYIiP5nZxl6NCh1CUYg7PQxzNPiZWVlbX6dWtS0vDhw5GQkOCLkkjMng1ceWURdRnGyM7Opi7BGJyFPt5og1jnzp1b/NrBgwexf/9+AI7eeiAPw/zqV0B5Oc9ItvTp04e6BGNwFvpIeuwNDQ0UpzVSa7NwrYumQGBOSrpQaWkpdQnG4CwUzkIfSdc5EO/DdlVL716klM5hmKSkJAwfPtyXZfnclClAfX1ffP01dSVm4OtQCmehj1tYQ+3fvx+HDh0CEPgXTRljnsUbbRCzluG9kNVbBxyzTTsC/rlQeKE8hbPQR9KwB9rKhO5o7m2mlNI5vj527FgMGjTI12WR4IvqSnR0NHUJxuAs9JE07C31Ujui5rb92r17N44fPw6gY1w0tfBtsEpbE9c6Es5CH0kXKSwsjOK0RurWrVuTf+uIwzALFwI2G2+NZxkwYAB1CcbgLPTxRhvEiorOn5Rjt9uds00nTZqE2NhYirJ87u67galTM6nLMMbhw4epSzAGZ6GPpGHnjTaUCydffP3118jNzQXgo2GYefOAyy8H4uOB117z/vlaUFUFXHJJ4M6s1RXIs4x1cRb6eNleYhcu22tdNA0KCsKCBQu8X8A//gHs3Qvs2QO88AJQRDOtf+ZMYNIkXrbXwkvVKpyFPt5og1jjZXttNhvWrFkDAJg6dapvplK/8AKwfr3j85MngSNHgDYWJvMWXrZX4aVqFc5CH/fYiTXusW/fvt25ubVPJiVt3w5s3Qrs3AmkpQGXXQYQXv84e5Z77BbupSqchT7eaINY4x67NQwTEhKC+fPne//kZWVA9+5AVBSQmQl88433z9kK7rEr3EtVOAt9vNEGsfz8fACOe7jXrVsHALj22msRExPj/ZNfdx1gswGXXgr8+tfAuHHeP2crmrunv6NKS0ujLsEYnIU+kjF2XtRH6d27NwBg69atKCkpAeDDtWHCw2HK7tFLlwINDXy3lCU+Pp66BGNwFvpIeuy1tbUUpzVScXExADUpKTw8HHPnzqUsicTSpcCECUeoyzDG0aNHqUswBmehj6Rh55mnSteuXVFTU4MNGzYAAGbOnImuXbsSV+V7hYVARETHmIzVHh1lYlp7cBb6eK0YYlVVVdi8ebPzjpCOukTvggXAokW8CJilsLCQugRjcBb6SBp23mhDCQsLcw7DREVFYdasWcQV0eFVP5XWtkzsaDgLfSQtrJSS4rRGqqurw6ZNmwAAP/nJT9CpUyfiiuhIyeuxW+rr66lLMAZnoY+7zsRyc3OdE7Y60hK9zeHf9wpvOqJwFvp4KIZYVlYWAMdF1Ouuu464GlrBwfxzYeFJfApnoY8vnhIqKytz9tavv/76Dn1//7JlwMKFNAuQmci6DZZxFq7wyG0IQojrAPwZQDCAv0sp/9Da40NDeUMFANiwYQOys7MB8DDMjTcC1dVNNx3pqBovNdHRcRb63O6xCyGCAbwMYAaAEQAWCyFGtPYc3gLN4Z133sHgwYMRExODadOmUZdD6uRJYMeOk9RlGMPaGpFxFq7wRI99LICjUspjACCEeBvAXAAZLT2hIw85WIqKirBlyxbYbDbccccdHf5dzC23AMAQXHMNdSVmGD58OHUJxuAs9HmiYe8PoHFXKwfAFa09gZftdQxHPffcc3jssUnYtetHmDJFfW3hQsdWcVVVjg0oLrR0qeOjsNAxsedCy5Y5hjZOnrQazPM98AAwZw5w6BBw111Nv/6b3wDTpgGpqcDy5U2//tRTwJVXAjt2AA8/3PTrK1cCycmOFYGffLLp1199FRg2DHj/feBPf3L8W2oqMHhwBQBe4REAUlNTMWrUKOoyjMBZ6PPZVD8hxJ0A7gSA6OhorFixAg0NDZBSIjQ0FNXV1ejSpQtKS0vRs2dP5Ofno1+/fjh16hQuuugi5OXloW/fvigsLER0dDTOnj2LyMhI1NfXQwiB4OBg1NXVISoqCuXl5YiJiUFBQQH69u3rPIb15+nTp9GjRw+UlZWhc+fOqKmpQUiIIwqbzYaIiAhUVFSgW7duKCoqQp8+fZocIz8/H71790ZxcTG6du2KqqoqhIWFab8mKSfizJkzKCkpQUNDA4KCgrBu3XfIytoJuz0CR4/+HMHBQbDZGhAaGoK6unps2JCBHTs+RHT0YBw+PBdhYeGoq6tDaGgobDYb3nprN775ZjeqqnogK2sxpJQICgpCQ0MDQkJC8Morm7F3bwEyMhpw4sRtqKurdR4jLCwUL764Cdu25eGHH7rj2LH5znkHQgjY7Xa8+uo2/Oc/+1FdfRkOH5507rm1zj//8pdtCAn5HiUll+Po0UkICgo67xhPPfUO+vevQFraAJw48WPU1dUjOjoMkZFf4oknsoz8PlH87P3tb38LuNfk6vfpnXfeCbjX5Mr3qd3trbuThYQQ4wE8IaX88bm/PwQAUsqnW3rOiBEjZEZGiyM1Hcqdd96J1wj3GjXJ3r17ee3tczgLhbNQhBB7pZSj23qcJ2533A1giBAiTggRBmARgPdaewLfl6rwFX+F//MqnIXCWehzu2GXUtoA/ALAxwAOAlgtpTzQ2nN4jF3Jy8ujLsEY+/btoy7BGJyFwlno88gYu5TyQwAftvfx3GNX+vbtS12CMZKTk6lLMAZnoXAW+khmntYQbphsGl6SVMnMzKQuwRichcJZ6OONNojpXOkOdHFxcdQlGIOzUDgLfSQNOy/DqVgbbDDg1KlT1CUYg7NQOAt9JA27dT8oAyIjzd/A+YsvgFGjgJAQYO1a750nJibGewf3M5yFwlnoI2nYeX1lxR/evQwYAKxaBdx0k3fPw3dLKZyFwlno4wWwiQkhPHq83buBxESgpgaorATi44Hvv3fvmAMHOo7p7WX0eZ1+hbNQOAt9JGMinm7M/Jmn9/kcMwb4yU8c671UVwM33wyMHNn0cZMmAc0N7z/7rGOdGAodfSG0xjgLhbPQR9Kw81CM4o0ljB97zNHAR0QAL7zQ/GO+/NLjp3VbRUUFevbsSV2GETgLhbPQR9Kw88VTxRuTtYqKgIoKoL7eMSTT3P7YJvbY+T+vwlkonIU+khaWN9pQysvLPX7Mu+4Cfv974Phx4H//F3jppaaPMbHHnpOTw2tvn8NZKJyFPpKrEuHh4RSnNZKnb+V6800gNNRxB8uvf+24mLptm3vH3L0biI0F1qxx/NKIj/dMrRcaPHiwdw7shzgLhbPQx0sKECsoKPDo8X72M2DdOsfnwcHAt98CV1/t3jHHjAFychx32RQVAQdaXeLNdQe8dWA/xFkonIU+kobdHybl+AovAqYkJSVRl2AMzkLhLPSRNOw84UDh6dLK3r17qUswBmehcBb6SBp2XrZX4Y02FN5QQeEsFM5CH/fYiXGPXeGemcJZKJyFPu6xE+Meu8I9M4WzUDgLfSQNe3V1NcVpjXT69GnqEoyRnp5OXYIxOAuFs9BH0rBHRERQnNZIPXr0oC7BGEOHDqUuwRichcJZ6CNp2HnmqVJWVkZdgjGys7OpSzAGZ6FwFvp4ow1inTt3pi7BGH369KEuwRichcJZ6CNp2BsaGihOaySehauUlpZSl2AMzkLhLPSRNOy8cL7C714UvvaicBYKZ6GPW1jGGAswvOcpMZvNRl2CMXhYSuEsFM5CH0nD7unt4PwZv81UoqOjqUswBmehcBb6SBp27qUqFRUV1CUYgydrKZyFwlnoI2nYw8LCKE5rpG7dulGXYIwBAwZQl2AMzkLhLPTxRhvEioqKqEswxuHDh6lLMAZnoXAW+nijDWI8+UJJSEigLsEYnIXCWejjZXuJ8bK9Ci/PqnAWCmehj5ftJcbL9iq8PKvCWSichT7usRPjHrvCPTOFs1A4C33cYyfGPXaFe2YKZ6FwFvp4ow1i+fn51CUYIy0tjboEY3AWCmehjzfaINa7d2/qEowRHx9PXYIxOAuFs9BH0rDX1tZSnNZIxcXF1CUY4+jRo9QlGIOzUDgLfTzzlFjXrl2pSzBGbGwsdQnG4CwUzkIfrxVDjO8QUgoLC6lLMAZnoXAW+nijDWL87kXhbQIVzkLhLPSRtLBSSorTGom3CVTq6+upSzAGZ6FwFvq460yMf8kpvAGLwlkonIU+HoohFhoaSl2CMXjimsJZKJyFPr54Sownayl866fCWSichT6Shp17qUqXLl2oSzAGL6+gcBYKZ6GPpGGvq6ujOK2RSktLqUswxvHjx6lLMAZnoXAW+nhJAWI9e/akLsEYw4cPpy7BGJyFwlno42V7ifEiYEpqaip1CcbgLBTOQh8v20usX79+1CUYY9SoUdQlGIOzUDgLfdxjJ8YbbSi8oYLCWSichT7usRPjK/4Kb6igcBYKZ6GPe+zE8vLyqEswxr59+6hLMAZnoXAW+rjHTqxv377UJRgjOTmZugRjcBYKZ6GPpGGvqamhOK2ReElSJTMzk7oEY3AWCmehjzfaIBYdHU1dgjHi4uKoSzAGZ6FwFvpIGnZehlM5e/YsdQnG4DuEFM5C4Sz0udWwCyFShBAHhBB2IcTo9j4vJCTEndMGlMjISOoSjBETE0NdgjE4C4Wz0Oduj/17APMBfKHzJF5fWeF3LwrfLaVwFgpnoc+trrOU8iAACCE8U00HxNkpvE6/wlkonIU+ksS4MVOCg4OpSzAGL+escBYKZ6FPtLU1mxBiK4DmbrZ+REq58dxjtgP4lZRyTyvHuRPAnef+OhKOYRwG9ATA9zw6cBYKZ6FwFsowKWWbmzi0ORQjpZzmiWqklK8BeA0AhBB7pJTtvtgayDgLhbNQOAuFs1CEEC12nhvjwSvGGAsw7t7ueL0QIgfAeAAfCCE+9kxZjDHGXOXuXTHrAax34amvuXPeAMNZKJyFwlkonIXSrizavHjKGGPMv/AYO2OMBRiyht3V5QgChRDiOiHEISHEUSHEr6nroSSE+IcQokAI0aFvgRVCXCyE+EwIkXHu/8a91DVREUJECCF2CSHSzmXxW+qaqAkhgoUQ3wkhNrX1WMoeu0vLEQQCIUQwgJcBzAAwAsBiIcQI2qpIrQJwHXURBrABeEBKOQLAOAD3dOCfi1oAV0spkwAkA7hOCDGOuCZq9wI42J4HkjXsUsqDUspDVOcnNhbAUSnlMSllHYC3AcwlromMlPILAMXUdVCTUuZJKfed+/wsHP+J+9NWRUM6VJz7a+i5jw57QVAIEQtgFoC/t+fxPMZOoz+Ak43+noMO+h+YNU8IMRDAZQC+pa2Ezrmhh1QABQC2SCk7bBYAVgJYAaBdKyh6tWEXQmwVQnzfzEeH7Z0y1hYhRGcA6wAsl1KWU9dDRUrZIKVMBhALYKwQYiR1TRSEELMBFEgp97b3OV5dGN1TyxEEoFwAFzf6e+y5f2MdnBAiFI5G/T9Synep6zGBlLJUCPEZHNdhOuIF9gkAfiKEmAkgAkBXIcS/pZQ3t/QEHoqhsRvAECFEnBAiDMAiAO8R18SICceyp68DOCilfI66HkpCiF5CiOhzn0cCmA6gQ25+KqV8SEoZK6UcCEdbsa21Rh2gvd2xwy5HIKW0AfgFgI/huEC2Wkp5gLYqOkKItwDsBDBMCJEjhLiNuiYiEwDcAuBqIUTquY+Z1EUR6QfgMyHEfjg6QluklG3e5scceOYpY4wFGB6KYYyxAMMNO2OMBRhu2BljLMBww84YYwGGG3bGGAsw3LAzxliA4YadMcYCDDfsjDEWYP4fcPdxfhqLc94AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(6, 6))\n",
    "axhline(y=0, color='k', linewidth=0.5)\n",
    "axvline(x=0, color='k', linewidth=0.5)\n",
    "grid(linestyle='dotted')\n",
    "axis([-1, 4, -1, 4])\n",
    "annotate('Vector [1, 3] representing x = 1, y = 3', xy=(0.5, 3.1), ha='left')\n",
    "quiver([0], [0], 1, 3, angles='xy', scale_units='xy', scale=1)\n",
    "plot([0, 1, 1, 1], [0, 0, 0, 3], 'b--')\n",
    "annotate('x = 1', xy=(0.5, -0.25), ha='center', color='blue')\n",
    "annotate('y = 3', xy=(1.1, 1.0), ha='left', color='blue')\n",
    "annotate('a', xy=(0.5, 0.2), ha='center', color='red')\n",
    "annotate('b', xy=(0.9, 1.5), ha='right', color='red')\n",
    "annotate('c', xy=(0.3, 1.5), ha='right', color='red')\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's easy to see that the length of $\\vec{v}$ is simply the hypotenuse of the right triangle formed with the $x$ and $y$ components of $\\vec{v}$. We can use the Pythagorean theorem to find the length of $\\vec{v}$ (represented as $c$ in the above visualization). We allow the $x$ leg to be $a$, and the $y$ leg to be $b$:\n",
    "\n",
    "$$\n",
    "a^2 + b^2 = c^2 \\\\[3pt]\n",
    "1^2 + 3^2 = c^2 \\\\[3pt]\n",
    "10 = c^2 \\\\[3pt]\n",
    "c = length(\\vec{v}) = \\sqrt{10}\n",
    "$$\n",
    "\n",
    "The Pythagorean theorem can also be rewritten as:\n",
    "\n",
    "$$c = \\sqrt{a^2 + b^2}$$\n",
    "\n",
    "Naturally, we arrive at the same result:\n",
    "\n",
    "$$\n",
    "c = \\sqrt{1^2 + 2^2} \\\\[3pt]\n",
    "c = \\sqrt{10}\n",
    "$$\n",
    "\n",
    "Fortunately, the process of finding the magnitude applies to vectors of any size and located anywhere in the plane because we can always form a right angle. Formally, given a vector $\\vec{v}$ of arbitrary size, we can find its magnitude using the same process:\n",
    "\n",
    "$$\n",
    "\\vec{v} = \\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\dots \\\\ v_i\\end{bmatrix} \\hspace 3em\n",
    "length(\\vec{v}) = \\sqrt{(v_1)^2 + (v_2)^2 + \\dots + (v_i)^2}\n",
    "$$\n",
    "\n",
    "It's commonplace to represent the magnitude of a vector as the vector surrounded by double bars, $\\left\\Vert\\vec{v}\\right\\Vert$:\n",
    "\n",
    "$$\\vec{v} = \\begin{bmatrix}1 \\\\ 3\\end{bmatrix} \\hspace 3em length(\\vec{v}) = \\left\\Vert\\vec{v}\\right\\Vert = \\sqrt{10}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Understanding magnitude, let's see how we might implement the length/magnitude of a vector. To illustrate, we can use the vector $\\vec{v}$ from above:\n",
    "\n",
    "$$\\vec{v} = \\begin{bmatrix}1 \\\\ 3\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Magnitude of [1, 3] = 3.162278\n",
      "Magnitude of [1, 2, 3, 4] = 5.477226\n",
      "Magnitude of [3, 4] = 5\n"
     ]
    }
   ],
   "source": [
    "from math import sqrt\n",
    "\n",
    "# Find the length of a vector: magnitude = sqrt(v1^2 + v2^2 + ... + vi^2)\n",
    "def magnitude(vector):\n",
    "    # First, square all the components, then sum the squares and take the square root\n",
    "    squares = [ comp**2 for comp in vector ]\n",
    "    return sqrt(sum(squares))\n",
    "\n",
    "v = [1, 3]\n",
    "v_length = magnitude(v)   # > sqrt(5) = 3.162...\n",
    "\n",
    "# Magnitude can work on vectors with more than 2 components\n",
    "a4 = [1, 2, 3, 4]\n",
    "a4_length = magnitude(a4) # > sqrt(30) = 5.477...\n",
    "\n",
    "other = [3, 4]\n",
    "other_length = magnitude(other)  # > 5\n",
    "\n",
    "print('Magnitude of %s = %f' % (v_to_str(v), v_length))\n",
    "print('Magnitude of %s = %f' % (v_to_str(a4), a4_length))\n",
    "print('Magnitude of %s = %d' % (v_to_str(other), other_length))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Component and standard form\n",
    "\n",
    "When we draw vectors geometrically using the components to represent axes in the coordinate system (like we've been doing), this is known as drawing a vector in **component form**. Often times, for convenience sake, vectors will be drawn starting from the origin. In this case, vectors are said to be in **standard form**. Drawing a vector in _standard form_ means the vector is drawn as a directed line segment from the origin to the point that corresponds to the vector's components."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Positions of vectors\n",
    "\n",
    "All of the vectors drawn above are drawn in _standard form_. This is a matter of convenience. There are no rules that require vectors to be drawn in standard form. The 2 important characteristics of vectors are that they have a certain magnitude and a certain direction. In fact, the following two vectors representing $\\begin{bmatrix}1 \\\\\\ 2\\end{bmatrix}$ are identical:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "hideCode": true,
    "scrolled": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x432 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "figure(figsize=(6, 6))\n",
    "axis([0, 5, 0, 5])\n",
    "grid(linestyle='dotted')\n",
    "quiver([0], [0], 1, 2, angles='xy', scale_units='xy', scale=1, color='red')\n",
    "quiver([2], [2], 1, 2, angles='xy', scale_units='xy', scale=1, color='blue')\n",
    "annotate('Vector: [1, 2]', xy=(0.8, 1), xytext=(0.8, 1), ha='left', color='red')\n",
    "annotate('Vector: [1, 2]', xy=(0.8, 1), xytext=(2.8, 3), ha='left', color='blue')\n",
    "show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "These 2 vectors have the same exact direction and the same magnitude. One just happens to be in standard form (the red one) and the other \"starts\" from $x = 2$, $y = 2$, but they are fundamentally the same vector:\n",
    "\n",
    "$$\\begin{bmatrix}1 \\\\ 2\\end{bmatrix}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Vector dimensions\n",
    "\n",
    "Often, when discussing vectors, you'll hear mention of the word _dimension_. **Dimension** refers to how many components are in the vector. For example, the following vector:\n",
    "\n",
    "$$\\begin{bmatrix}3 \\\\ 1\\end{bmatrix}$$\n",
    "\n",
    "Is 2-dimensional. You'll often see it written as saying $\\begin{bmatrix}3 \\\\\\ 1\\end{bmatrix}$ is a member of $\\mathbb{R}^2$. This stylized \"R\" represents real numbers and the exponent of 2 simply means vectors with 2 real numbers as components. Here are a few other examples and their associated dimension:\n",
    "\n",
    "$$\n",
    "\\begin{bmatrix}3 \\\\ 1 \\\\ 2\\end{bmatrix} \\enspace\\rightarrow\\enspace \\mathbb{R}^3 \\qquad\n",
    "\\begin{bmatrix}1\\end{bmatrix} \\enspace\\rightarrow\\enspace \\mathbb{R}^1 \\qquad\n",
    "\\begin{bmatrix}3 \\\\ 1 \\\\ 2 \\\\ -2\\end{bmatrix} \\enspace\\rightarrow\\enspace \\mathbb{R}^4 \\qquad\n",
    "\\begin{bmatrix}5 \\\\ 0\\end{bmatrix} \\enspace\\rightarrow\\enspace \\mathbb{R}^2 \\qquad\n",
    "\\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\vdots \\\\ v_n \\end{bmatrix} \\enspace\\rightarrow\\enspace \\mathbb{R}^n\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Putting together all the code\n",
    "\n",
    "Below is a listing of the code we've implemented in this guide."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Size of [1, 2] = 2\n",
      "[1, 2] + [3, 4] = [4, 6]\n",
      "[1, 2] - [3, 4] = [-2, -2]\n",
      "3 * [1, 2] = [3, 6]\n",
      "3 * [1, 2] = [3, 6]\n",
      "Magnitude of [1, 2] = 2.236068\n",
      "[1, 2, 3, 4] + [9, 8, 7, 6] = [10, 10, 10, 10]\n",
      "[1, 2, 3, 4] - [9, 8, 7, 6] = [-8, -6, -4, -2]\n",
      "3 * [1, 2, 3, 4] = [3, 6, 9, 12]\n"
     ]
    }
   ],
   "source": [
    "a = [1, 2]\n",
    "b = [3, 4]\n",
    "\n",
    "# Returns the number of components in a vector\n",
    "def size(v):\n",
    "    return len(v)\n",
    "\n",
    "# Recall, in order to do component wise operations, the vectors need to be the same size\n",
    "def check_size_compatible(v1, v2):\n",
    "    if size(v1) != size(v2):\n",
    "        raise ValueError('Vectors are different sizes')\n",
    "\n",
    "# Adding 2 vectors component by component\n",
    "def add(v1, v2):\n",
    "    check_size_compatible(v1, v2)\n",
    "    return [v[0] + v[1] for v in zip(v1, v2)]\n",
    "\n",
    "# Subtracting 2 vectors component by component\n",
    "def subtract(v1, v2):\n",
    "    check_size_compatible(v1, v2)\n",
    "    return [v[0] - v[1] for v in zip(v1, v2)]\n",
    "\n",
    "# Multiply a vector and a number\n",
    "def multiply(v1, num):\n",
    "    return [num * v for v in v1]\n",
    "\n",
    "# Multiply a vector and a number using repeated addition (using functools.reduce)\n",
    "def multiply_alt(v1, num):\n",
    "    return reduce(add, [v1] * num)\n",
    "\n",
    "# Find the length of a vector using Euclidian distance\n",
    "def magnitude(vector):\n",
    "    squares = [ comp**2 for comp in vector ]\n",
    "    return sqrt(sum(squares))\n",
    "\n",
    "\n",
    "# Gets the number of components in a\n",
    "a_size = size(a)\n",
    "\n",
    "# Adds a and b\n",
    "a_plus_b = add(a, b)\n",
    "\n",
    "# Subtracts a and b\n",
    "a_minus_b = subtract(a, b)\n",
    "\n",
    "# Multiplies a by the scalar 3\n",
    "a_times_3 = multiply(a, 3)\n",
    "\n",
    "# Multiplies a by the scalar 3 using repeated addition\n",
    "a_times_3_repeated_addition = multiply_alt(a, 3)\n",
    "\n",
    "# The magnitude of the vector\n",
    "a_length = magnitude(a)\n",
    "\n",
    "print('Size of %s = %d' % (v_to_str(a), a_size))\n",
    "print('%s + %s = %s' % (v_to_str(a), v_to_str(b), v_to_str(a_plus_b)))\n",
    "print('%s - %s = %s' % (v_to_str(a), v_to_str(b), v_to_str(a_minus_b)))\n",
    "print('%s * %s = %s' % ('3', v_to_str(a), v_to_str(a_times_3)))\n",
    "print('%s * %s = %s' % ('3', v_to_str(a), v_to_str(a_times_3_repeated_addition)))\n",
    "print('Magnitude of %s = %f' % (v_to_str(a), a_length))\n",
    "\n",
    "# Operations with vectors of size 4\n",
    "a4 = [1, 2, 3, 4]\n",
    "b4 = [9, 8, 7, 6]\n",
    "\n",
    "a4_plus_b4 = add(a4, b4)\n",
    "a4_minus_b4 = subtract(a4, b4)\n",
    "a4_times_3 = multiply(a4, 3)\n",
    "\n",
    "print('%s + %s = %s' % (v_to_str(a4), v_to_str(b4), v_to_str(a4_plus_b4)))\n",
    "print('%s - %s = %s' % (v_to_str(a4), v_to_str(b4), v_to_str(a4_minus_b4)))\n",
    "print('%s * %s = %s' % ('3', v_to_str(a4), v_to_str(a4_times_3)))"
   ]
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    "## Summary, tl;dr\n",
    "\n",
    "In this guide, we introduced the vector, which is one of the basic primitives of linear algebra. We explored what a vector is, what a vector represents, how to perform operations on vectors, and how to represent vectors.\n",
    "\n",
    "### Vectors\n",
    "\n",
    "Vectors are defined as a list of numbers surrounded by square brackets:\n",
    "\n",
    "$$\\vec{v} = \\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\vdots \\\\ v_n\\end{bmatrix}$$\n",
    "\n",
    "Where $v_1, v_2, \\dots v_n$ are called the components of the vector. Vectors are represented as a symbol with an arrow over it: $\\vec{v}$.\n",
    "\n",
    "### Magnitude and direction\n",
    "Vectors have a magnitude and direction which uniquely define a vector. The magnitude corresponds to _how long_ a vector is, and the direction corresponds to _which direction_ the vector is pointing in. The magnitude is referenced as $\\left\\Vert\\vec{v}\\right\\Vert$ and is calculated using the Pythagorean theorem:\n",
    "\n",
    "$$\\left\\Vert\\vec{v}\\right\\Vert = \\sqrt{(v_1)^2 + (v_2)^2 + \\dots + (v_i)^2}$$\n",
    "\n",
    "### Operations\n",
    "Vectors behave _similar_ to real numbers in that they can be added, subtracted, and multiplied by a scalar number. These operations are performed component wise. The result of these vector operation is another vector. For example, vector addition:\n",
    "\n",
    "$$\n",
    "\\vec{v} = \\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\ldots \\\\ v_n\\end{bmatrix}, \\vec{w} = \\begin{bmatrix}w_1 \\\\ w_2 \\\\ \\ldots \\\\ w_n\\end{bmatrix}, \\enspace\\enspace\n",
    "\\vec{v} + \\vec{w} = \\begin{bmatrix}v_1 + w_1 \\\\ v_2 + w2 \\\\ \\ldots \\\\ v_n + w_n\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "$\\vec{v}$ and $\\vec{w}$ must have the same number of components. Same for subtraction:\n",
    "\n",
    "$$\\vec{v} - \\vec{w} = \\begin{bmatrix}v_1 - w_1 \\\\ v_2 - w2 \\\\ \\ldots \\\\ v_n - w_n\\end{bmatrix}$$\n",
    "\n",
    "Scalar multiplication by a number $c$ is done by multiplying each component by $c$:\n",
    "\n",
    "$$\n",
    "\\vec{v} = \\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\ldots \\\\ v_n\\end{bmatrix}, \\enspace\\enspace c\\vec{v} = \\begin{bmatrix}cv_1 \\\\ cv_2 \\\\ \\ldots \\\\ cv_n\\end{bmatrix}\n",
    "$$\n",
    "\n",
    "In subsequent documents, we'll dig deeper into more core concepts and build on our knowledge of vectors.\n",
    "\n",
    "### Geometry\n",
    "A vector can be viewed geometrically as a directed line segment in space with a starting point and an ending point. When visualizing vectors geometrically, vectors are location agnostic in that as long as the magnitude and direction are the same, it doesn't matter where they are plotted in the plane.\n",
    "\n",
    "### Linear combination\n",
    "A combination of scalar multiplication and addition is a linear combination. Consider a linear combination of scalar numbers $c$ and $d$ and vectors $\\vec{v}$ and $\\vec{w}$:\n",
    "\n",
    "$$\n",
    "\\vec{v} = \\begin{bmatrix}v_1 \\\\ v_2 \\\\ \\ldots \\\\ v_n\\end{bmatrix}, \\vec{w} = \\begin{bmatrix}w_1 \\\\ w_2 \\\\ \\ldots \\\\ w_n\\end{bmatrix}, \\enspace\\enspace\n",
    "c\\vec{v} + d\\vec{w} = \\begin{bmatrix}cv_1 + dw_1 \\\\ cv_2 + dw_2 \\\\ \\ldots \\\\ cv_n + dw_n\\end{bmatrix}\n",
    "$$"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## What's next?\n",
    "\n",
    "In the next guide, we'll be introduced to [matrices](matrices) which are the next logical progression after vectors. What you'll see is that matrices are very similar to vectors."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<nav class=\"links\">\n",
    "    <a href=\"introduction\" class=\"prev\">Introduction</a>\n",
    "    <a href=\"matrices\" class=\"next\">Matrices</a>\n",
    "</nav>"
   ]
  }
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