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# quantgirluk/Brownian Motion Random Walks.ipynb

Created January 20, 2019 15:20
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 { "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Brownian Motion via Random Walks\n", "\n", "January 2019 @Quant_Girl\n", "\n", "The idea of this post is to show how Donsker's Theorem (also known as Donsker's invariance principle, or the functional central limit theorem) allows us to simulate paths of the one-dimensional standard Brownian motion using different kinds of random walks.\n" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [], "source": [ "import sys\n", "import math\n", "import random\n", "import pandas as pd\n", "import numpy as np\n", "import matplotlib.pyplot as plt\n", "import matplotlib.animation as animation\n", "from matplotlib import style\n", "from PIL import Image\n", "from IPython.display import HTML\n", "import seaborn as sns\n", "sns.set(style=\"whitegrid\")" ] }, { "cell_type": "code", "execution_count": 2, "metadata": {}, "outputs": [], "source": [ "%matplotlib inline\n", "plt.rcParams['figure.figsize'] = [14, 7]\n", "np.random.seed(123)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Simple Random Walk\n", "\n", "First, we simulate a simple random walk." ] }, { "cell_type": "code", "execution_count": 8, "metadata": {}, "outputs": [], "source": [ "def simple_walk(x0=0, n=30, step=1, p=0.5, vis=True):\n", " \n", " '''\n", " Returns a list with the values of a simple Random Walk defined by\n", " the following parameters.\n", " \n", " x0 : starting point\n", " n : lenght of the walk\n", " step : size of the steps\n", " p : probability of going up\n", " vis : to display de plot True\n", " \n", " '''\n", " \n", " values = [x0] + [step if random.random()>p else -step for i in range(1,n)]\n", " walk = np.cumsum(values)\n", "\n", " if vis == True: \n", " plt.plot(walk, linestyle='--', marker='o', color=\"purple\")\n", " plt.axhline(lw=0.5, color='black')\n", " plt.ylabel('$S_k$', fontsize=14)\n", " plt.xlabel('k', fontsize=14)\n", " plt.title('Simple Random Walk n='+ str(n), fontsize=16)\n", " plt.show()\n", " \n", " return walk" ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [ { "data": { "image/png": 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ZKopERET8zJpX17Bt7jYGvzCYplc0rdA2RflFvNv3XaaNnUZJcYmPE4qIBBcVRSIiIn4m51AOl914GT0e7FHhbcKjwhk4YSC7l+1m5T9W+jCdiEjw0TVFIiIifmbAXwZQUlyCYRiV2q7rbV3ZNmcbi59aTIdBHWjZvaWPEoqIBBf1FImIiPiJFc+vYPcye6a5MFflX6INw2DEGyOoe0ldUlNSKcgp8HZEEZGgpKJIRETED+xI38GiJxex8aON1dpPdMNoRk8aTdt+bb2UTEQk+Gn4nIiIiMPyjuYx/fbpNLm8CUNeGFLt/bUf0J72A9p7IZmISGhQT5GIiIiD3G43s+6bRc6hHJKnJhNRO8Jr+z6YcZD3+r3HyQMnvbZPEZFgpKJIRETEQVvnbGXzp5sZ8NcBtOjWwqv7DgsPY9/qfcy4cwZut9ur+xYRCSYqikRERBzUcXhHkqcm0/uJ3l7fd9NOTRny4hC2zdvG6v+s9vr+RUSChYoiERERB5QUlXBi/wmMMIO4W+KqNNtcRXR/oDsdR3Rk4W8WcmjTIZ+0ISIS6FQUiYiIOGDZ35bx2pWvceKHEz5txzAMRr49kqj6UXz5wpc+bUtEJFBp9jkREZEatnflXpb9dRlxKXHUa1nP5+3VbV6XO5beQaNLG/m8LRGRQKSeIhERkRqUn51P6thUYtrGcMN/bqixdpt2aoorwkXukVz2r9tfY+2KiAQCFUUiIiI1aO4v55K1O4ukyUlE1Y+q8fY/velTPkz8kNwjuTXetoiIv1JRJCIiUkOKC4spLiimz/g+xF4X60iGwc8PJudwDrPGzdI03SIiHiqKREREaogrwsWYD8bQ/0/9HcvQolsLBv5tIN+lfsf6d9c7lkNExJ+oKBIREfExd4mbOQ/P4fB3hwEwwgxH81z7+LW069+Oub+cy9FtRx3NIiLiD1QUiYiI+NjKF1ey5tU1ZH6Z6XQUAMJcYYyeNBoz0SSyXqTTcUREHKcpuUVERHxo/zf7+Xz853RK7sRVd17ldJzTYtrEMObDMQC43W4Mw9neKxERJ6mnSERExEcKcwtJTUmlTtM63DjxRr8sPE7sP8GkgZPYs2KP01FERByjokhERMRHvnzpS3787kdGvz+a2o1rOx3nnCLrRHJ893GmjZ1Gfna+03FERByhokhERMRHej/em5um30SHQR2cjnJeUfWjSJqcRNaeLOY+MtfpOCIijnCkKDJNc6xpmps8Py84kUFERMRXco/kkp+dT3itcC4fdbnTcS4q9rpY+vyhDxsmbWDTJ5ucjiMiUuNqfKIF0zRrA/8CLgOOAytM0xxkWdaims4iIiLiTRlTM0j/fTpZe7IICw9j5Dsj6XprV6djVUi/p/qxff52Fj65kIW/WUjW3ixiYmNImJBAXEqc0/FERHzKiZ4il6fdOkCE5yfPgRwiIiJekzE1g7RxaWTtyQKgpKiE2ffPJmNqhsPJKiYsPIy4lDhyDubYx+CGrN1ZpI1LC5hjEBGpqhoviizLOgE8BXwPZAK7gJU1nUNERMSb0senU5hbeMaywtxC0senO5So8r588UuK8orOWBZoxyAiUhWG2+2u0QZN0+wCvA8MBbKAKcBqy7L+caHt1q5d2w7Y6fOAIiIiVTCrxyw410uqATeuubHG81RFMByDiEgZ7ePj43dVZEUnvrx1KJBuWdYhANM03wMeBC5YFJXq3LkzUVFRvktXCWvXriU+Pt7pGOJjOs+hQ+c6NPjqPC9rtYzszOxyy2NiYwLmcbU8djlZu7PKLQ+kYyhLf9OhQec5dFT0XOfn57Nx48ZK7duJa4o2AINM06xjmqYBJAJrHMghIiLiNYOeG0RE7YgzlkXUjiBhQoJDiSovYUJCuWMICw8LqGMQEakKJ64pWgB8CKwFvsWeaOG5ms4hIiLiDXnH8phx5wzaJ7QncWIiMW1jwICYtjEkTkwMqJnb4lLizjiGyHqRlBSVEBXjHyM0RER8xYnhc1iW9Xfg7060LSIi4i1ut5vZ98/mu9Tv6P5gd+JS4gKqCDqXssdQlF9E2r1pNLq0kcOpRER8y5GiSEREJBh8O/lbNn2yiYETBtKqRyun43hdeFQ4SZOSALsABDAMw8lIIiI+4cQ1RSIiIgHv2I5jzHl4DrF9YrnuyeucjuNThXmFfHbzZ6x5TZcAi0hwUlEkIiJSBYueXIRhGCRNTiLMFdwvp+G1wsnPymfhEws5vPmw03FERLwuuJ/FRUREfOTG/97IzWk306BtA6ej+JxhGIx6dxSRdSNJTUmlKL/o4huJiAQQFUUiIiKVcHzXcYoLi4luFE3bvm2djlNj6l5Sl5Fvj+TA+gMsfmqx03FERLxKRZGIiEgF5Z/IZ9KgSXx282dOR3GEOdIk/r541r+7nrxjeU7HERHxGhVFIiIiFTTvsXkc33mcno/2dDqKY4a8OIT7N9xPdMNop6OIiHiNiiIREZEK2PzZZta/s57r/u862vYJnWFzZ4usE0m9lvVwl7j5btp3p6fqFhEJZCqKRERELiJ7XzZp96bRsntL+v+pv9Nx/ELGBxl8kvwJG97f4HQUEZFqU1EkIiJyEfnZ+TTu2Jjkqcm4IlxOx/ELnW/uTNu+bZn7yFyObj/qdBwRkWpRUSQiInIRTTs15e5Vd9P4ssZOR/EbYa4wkiYnYbgMpt06jZKiEqcjiYhUmYoiERGR8ziw4QBp96VRcLIAwzCcjuN3YmJjGPH6CDK/zGTZhGVOxxERqTIVRSIiIudQmFdI6i2pbJm5haJT+rLS84m7OY7ev+1N+4HtnY4iIlJl4U4HEBER8UeLnlzE4c2HSZmXQu0mtZ2O49cG/33w6f+73W71qolIwFFPkYiIyFm2zt3K6n+vpuejPbl06KVOxwkYC59cyMx7ZjodQ0Sk0lQUiYiIlFFSXML8x+bTrHMzBj03yOk4AcUV4WL9O+vZ/Olmp6OIiFSKhs+JiIiUEeYKY+z8sRTmFRJeSy+TldHv6X5sX7CdtHFptO7Vmvqt6zsdSUSkQtRTJCIi4nF482HcbjcN2jWgaaemTscJOK4IF8lTkykuKGb6HdNxl7idjiQiUiEqikRERIAfrR+Z2H0iy/6qqaWro3HHxgx7eRh7V+zlwIYDTscREakQjQsQEZGQV1xQTGpKKhHREXS7p5vTcQLe1XdfTYfBHWjQtoHTUUREKkQ9RSIiEvIWP72Y/Wv3k/hWIvVa1nM6TsAzDON0QfT99O8pzCt0OJGIyIWpKBIRkZC2a+kuVvx9BVfffTWdkjo5HSeoHMw4yMdJH7PwtwudjiIickEqikREJKS5S9y07dOWYS8PczpK0Gke15yej/VkzX/WsHXuVqfjiIicl4oiEREJae0HtOf2JbcTWTfS6ShBadCzg2gW14wZd84g51CO03FERM5JRZGIiISkjA8yWPz0YkqKSjAMw+k4QSu8VjjJU5M5dfwUM++ZidutabpFxP+oKBIRkZBzfNdxZj8wm53pO0H1kM81j2vO8H8Np8utXVSAiohf0pTcIiISEjKmZpA+Pp2sPVm4Il0AJE1OIsylzwdrQvy4eMBzHn6fTtbeLGJiY0iYkEBcSpzD6Sqn7GMpUI9BRM6kokhERIJextQM0salUZhrTw1dnF+MK9JF5spMGrZv6HC60JExNYMZd82guKAYgKzdWaSNSwMImKLi7MdSIB6DiJSnj8dERCTopY9PP/0mtlRxQTHp49MdShSa0senny6IShXmFgbUeTjXYynQjkFEylNRJCIiQS9rT1allotvBMN5CIZjEJHyVBSJiEjQi4mNqdRy8Y1gOA8xbQL/GESkPBVFIiIS9C5PurzcsojaESRMSHAgTehKmJBARO2IM5a5Il0BdR5+MvQn5ZYZLiOgjkFEylNRJCIiQe3EDyf4dvK3xLSNsT/NNyCmbQyJExN1YXwNi0uJI3FiIjFt7fNQr1U9Rr0zKmDOw9HtR9n44UaadGpy+rEUFROFu9iNEa6pxkUCmWafExGRoOUucTPjzhkU5hZy1xd30eTyJqxdu5b4+Hino4WsuJS4ckXQrqW7aNC2AQ3aNXAo1cUVFxaTmpJKWHgYY+ePPT2MrqSohHeuf4cFjy+gU1Kn09O9i0hgUVEkIiJB66t/f8X2BdsZ8foImlzexOk4cg752fl8nPQxTa9oyh1L7iAs3D8HsSz72zL2fbWPMR+NOeO6orDwMMZ8MIaS4hIVRCIBzD+feURERLwgulE0nW/uTPx96hnyV1H1oxj+7+HsXbGXL577wuk459UyviU9H+tJ55s6l7utYYeGNO7YGLfbzaGNhxxIJyLVpaJIRESCVtdbuzLmgzEYhq738GddUroQd0scS/60hH2r9zkd5wxutxsAc6TJsJeGXXDdVS+v4r/d/sv+dftrIpqIeJGKIhERCTrp49NZ99Y6p2NIJdzw6g3Ub1Wf1JRUCnIKnI5z2qz7ZrHyhZUVWveq26+iTrM6fHbLZ+W+4FVE/JuKIhERCSrbF2zni2e+4MCGA05HkUqo1aAWSZOT6P5gdyKiIy6+QQ3Y9Mkm1r25jvzs/AqtH90omtHvj+aIdYQFTyzwcToR8SYVRSIiEjRyf8xl+h3TaXpFUwY/P9jpOFJJbfu25dpfXYsRZlBSXOJolqy9Wcy6bxatrmlF36f6Vni7DgkduPbxa/n69a/ZMmuLDxOKiDepKBIRkaDgdrtJuzeNvCN5JE9N9pveBqm87Qu382qnVznxwwlH2i8pLmH6bdMpLiwmeWoyrojKzSo3cMJAOgzugOHStWwigUJFkYiIBIXMLzP5fvr3DHxmIJdcdYnTcaQaYmJjOLHvBNPvmI67xF3j7WeuymT3st0M/9dwGl3aqNLbh0eFc+uCW+k4vKMP0omIL6goEhGRoNCmdxvu/OJOrv3VtU5HkWpqYjZh6EtD2bFwB6teWVXj7cdeF8sDGQ9w1Z1XVWs/7hI3y59dztdvfO2lZCLiKyqKREQkoBUXFnNgvT2pQux1sRhhGrIUDLrd2w1zlEn6/6Vz8NuDNdJmYW4hu5buAqDpFU2rP5W7AXuW72H+r+ZzePPh6gcUEZ9RUSQiIgFt6Z+X8maPNzmy5YjTUcSLDMMg8c1EohtFs/nTzTXS5oLfLGDSwEkc2eqdx5JhGIx6ZxSRdSNJTUmlKL/IK/sVEe9TUSQiIgFrzxd7+OLZL+hyaxcaX9bY6TjiZXWa1uG+b+5jwF8G+LytLbO28PVrX9Pz0Z407ui9x1LdS+oy8p2RHFh/gMVPLfbafkXEu1QUiYhIQDqVdYrUsak0aN+AYa8MczqO+EjdS+oCcHjzYXYv3+2TNk4ePMmMu2bQvEtzEp5J8Pr+zUST+PvjWfXyKrL2ZHl9/yJSfeFOBxAREamKuQ/PJTszm7u+uIuoelFOxxEfcrvdTL9jOtl7s7n/2/up07SOV/c9866Z5Gfnc/vi2wmv5Zu3RkNfHMrVd15NTGyMT/YvItWjniIREQk4brebxmZj+v+5P617tXY6jviYYRiMfGskeUfzSLs3Dbfbu9N0dxzRkWEvD6PZlc28ut+yImpH0OqaVgDsX7ff68cgItWjokhERAKOYRj0/UNf+o7v63QUqSHNuzQn4dkErBkW695a55V9ut1uDMOgx4M96H5/d6/s82J2LNrBxPiJbHh/Q420JyIVo6JIREQCRklxCZ/e9Cnb5m1zOoo4oNdjvegwqAPzH5tf7RniivKLmDRwUo3NbFeq3YB2tO3XlrmPzOXo9qM12raInJ+KIhERCRgr/7GSTZ9sIudQjtNRxAFGmMHo90fT87Ge1b42Z/FTi9m1ZBeuSJeX0lVMmCuMpElJGC6DabdOo6SopEbbF5FzU1EkIiIB4Yevf2DxU4u58udX0uXWLk7HEYfUa1mPhAkJhEeFU1xYXKV97Fy8k5UvrKTbuG6YI00vJ7y4mNgYbnzjRjK/zGTZhGU13r6IlKeiSERE/F5BTgGpKanUvaQuI94YgWEYTkcShx3+7jCvdnq10tN05x3NY9qt02jcsTFD/znUR+kurvMvOhN/Xzz1WtRzLIOI/I+KIhER8XsbJm3gyNYjjJ40muiG0U7HET8Q0yYGwzCYNnYap46fqvB2mz8DLVzwAAAgAElEQVTdTM7BHJI/SCayTqQPE17cjW/cSPy4eEcziIhNRZGIiPi97vd35+4v76b9gPZORxE/EVk3kqQpSWTvy2bOQ3MqvF38uHge3PwgLeNb+jBd5Xw75Vvm/nKu0zFEQpqKIhER8VsnD57k2I5jGIZB6576PiI5U+ueren3dD8yPsgg44OMC657bOcxDmw4AEDjjo1rIl6F/Wj9yOp/r2bT/9vkdBSRkKWiSERE/JLb7WbGHTN4+9q3KcwtdDqO+Kk+v+tDm+vasPn/nX9q7ZKiElJTUpkyZAqFef73WOr3x360uqYVs+6bRXZmttNxREKSiiIREfFLq/+zmm3zttHv6X5E1I5wOo74qbDwMG6eeTM/+/Rn511n+TPLyfwyk6EvDyUi2v8eS64IF8lTkykuKGb67dNxl7idjiQSclQUiYiI3zm06RALf7OQjiM60v2B7k7HET8X3SiaMFcYJ/afYNMnZw5By1yVydK/LCUuJY64m+McSnhxjS5txLBXhrHz851sX7Dd6TgiISfciUZN00wEngbqAAssy3rUiRwiIuJ/ivKLSL0llVoxtRj59khNvy0VtuRPS1j/znoadmhIy+4tKcotIvXOVOq3rs8Nr97gdLyLuvquq2naqSlterdxOopIyKnxosg0zQ7AG0BP4CDwuWmawy3L0rQrIiIhLGNqBunj08nak0VEnQi639edus3rOh1LAsig5waxbe42Pkj8AFeEi+zMbKLqRXHNo9dQK6aW0/EuyjCM0wXRF89+wZrX15CdmU1MbAwJExKIS/Hfni6RQOfE8Lkk4GPLsjItyyoEbgK+ciCHiIj4iYypGaSNSyNrdxa4ofBkIV+//jUZUy88o5hIWdENo+l6W1dyDuSQvTcb3JCfnc+qF1cF1GPpq39/Rfrv008fQ9buLNLGpQXUMYgEGieKoksBl2maM03TXA88CBxzIIeIiPiJ9PHp5WaYK8wtJH18ukOJJFB9O+XbcssC7bH05YtfllsWaMcgEmicuKYoHOgL9AdOAjOB24H3KrLxxo0bfZWrStauXet0BKkBOs+hQ+faGVl7ss673BfnROc5eNX0Y8kXguEYaprul9Dhq3PtRFF0AFhkWdZhANM0pwHXUMGiqHPnzkRFRfkuXSWsXbuW+Ph4p2OIj+k8hw6da+ekN0on70heueUxsTFePyc6z8FteexyexjmWXzxWPKVYDiGmqS/6dBR0XOdn59f6Y4UJ4bPzQKGmqbZwDRNFzAcUHkvIhKijm47Sv6JfIywM2eZi6gdQcKEBIdSSaBKmJBQ7nutAu2xFAzHIBJoarwosizrK+B54AtgM7AbeLemc4iIiH9Y+eJKImtHMvSlocS0jQEDYtrGkDgxUbNtSaXFpcSRODExoB9LwXAMIoHGke8psizrHeAdJ9oWERH/Mvxfw+nxYA+axzWn5y97Oh1HgkBcShxxKXEBPayq9BhK7Vqyiw9HfsjPP/05rkiXg8lEgpMTw+dEREQ4mHGQ3CO5uCJcNI9r7nQcEb92KusUW9K2sPiPi52OIhKUVBSJiEiNy8/O56ORH/Hx6I+djiISEC4fdTndxnVjxfMr2LVkl9NxRIKOiiIREalxcx+ZS9aeLAb9fZDTUUQCxtB/DqVxx8ZMu3UaecfKz9YoIlWnokhERGrUxo83smHSBvr8oQ9terdxOo5IwIisE0ny1GROHjjJujfXOR1HJKg4MtGCiIiEpqy9Wcy+fzate7Wm31P9nI4jEnBadm/J3avupkW3Fk5HEQkq6ikSEZEaEx4VTofBHUiakkRYuF6CRKqiZXxLDMMga08Wx3cfdzqOSFBQT5GIiNSYOs3q8LNPfuZ0DJGAV1xYzLt936V+q/rcsfQOfcggUk36CxIREZ/b/81+Jg2aRPa+bKejiAQFV4SLhGcS2LtyL8ufXe50HJGAp6JIRER8qjC3kNRbUvnx+x+JiI5wOo5I0Ii7JY64W+JY+uelZH6V6XQckYCmokhERHxqwRML+PH7Hxn9/miiG0U7HUckqNzw6g3Ub12f1JRU8k/kOx1HJGCpKBIREZ/ZMmsLX7/+Ndc+fi0dEjo4HUck6NRqUIukyUm07dPW6SgiAU0TLYiIiE+43W6WP7Oc5l2bM3DCQKfjiASttn3aqigSqSb1FImIiE8YhsHY+WP5xfRfEB6lz+BEfO3wd4d5f8D7mtBEpApUFImIiNftXr6bwrxCoupF0aBdA6fjiISEMFcY+1bvY8YdM3CXuJ2OIxJQVBSJiIhXHd58mClDprDwNwudjiISUhpf1pihLw9lx6IdrHp5ldNxRAKKiiIREfGaovwiUlNSiawbSd8/9HU6jkjI6XZPN8xRJum/S+fAhgNOxxEJGCqKRETEaxY/tZgD6w8w8p2R1L2krtNxREKOYRiMfGsk0Y2iWfn8SqfjiAQMXfkqIiJesfPznax8YSXx98VjJppOxxEJWbWb1Oa2z2+j0U8aOR1FJGCop0hERLyiXqt6dL6pM0NeHOJ0FJGQ17RTU1yRLk4dP8UPa39wOo6I31NRJCIi1eJ2u3G73TQxmzDmwzFE1ol0OpKIeKSmpPLBiA/IOZzjdBQRv6aiSEREqmXDpA189ovPKDhZ4HQUETlLwrMJnDp2irR70nC7NU23yPmoKBIRkSo7uv0ocx+ey8kDJwmP1mWqIv6meZfmJDyXgDXTYt2b65yOI+K3VBSJiEiVlBSVMO3WaRgug6TJSYS59JIi4o96PdqLDoM6MP9X8zmy5YjTcUT8kl7BRESkSpZNWEbml5mMeH0EMbExTscRkfMwwgxGvz+ajiM6EllX1/yJnIvGOoiISKXln8jn69e/psvYLsTdHOd0HBG5iHot6/GzT34G2JOjGIbhcCIR/6KeIhERqbSoelHc9819DP/PcKejiEgl5BzKYcrQKexettvpKCJ+RT1FIiJyURlTM0gfn07WnixqN6nN0H8OpcvYLk7HEpFKiqgdwfGdx/k46WMiakeQvS+bmNgYEiYkEJcSWL2+ZZ+XlscuD8hjEP+hniIREbmgjKkZpI1LI2t3Frgh93AuM++eScbUDKejiUglRdaNJG5sHHlH88jOzAY3ZO3OIm1cWkD9TZ/9vBSIxyD+RUWRiIhcUPr4dApzC89YVlxQTPr4dIcSiUh1rH93fbllhbmFAfU3fa7npUA7BvEvKopEROSCsvZkVWq5iPi3YPibDoZjEP+iokhERC7ofNNtaxpukcAUDH/TwXAM4l9UFImIyAUlTEjAFeU6Y1lE7QgSJiQ4lEhEqiNhQgIRtSPOWOaKcgXU33TChATCo8+cL8wVGVjHIP5FRZGIiJyX2+0mLiWOUW+PIqZtDBgQ0zaGxImJmuVJJEDFpcSRODHx9N90VP0oiguKaXRZI6ejVUhBTgGdb+nMyDdHnj6G6CbRJL6l5yWpOk3JLSIi5zXv0XmERYQx5IUherMhEkTiUuJO/03nHcvjja5vkJqSyn3r7iOybqTD6c7P7Xbz6c8/JbJuJGM+GkNcShxr164lPj4egNwfc4moHVGuJ0zkYtRTJCIi57R1zlZW/3s17hI3hmE4HUdEfCS6YTRJk5I4uu0o83893+k4F/T161+zdc5W2lzfptzz0qmsU7zR9Q0W/nahQ+kkkKkoEhGRcnIO5TDjzhk0i2vGoGcHOR1HRHysXf92XPfb61j35jq+n/6903HO6fB3h1nw+AIuHXYp1zx8Tbnba8XU4spfXMmaV9ewZfYWBxJKIFNRJCIiZ3C73cy8eyansk6RPDWZ8FoaaS0SCgb8ZQBdbu1Co0v979qi4oJiUlNSiawbyah3R5239zrhmQSad2nOzLtmknMop4ZTSiBTUSQiImc4vPkwOxbtYNBzg2ge19zpOCJSQ1yRLpImJdGsczPA/oDEXxzdfpST+08y8u2R1L2k7nnXC48KJ/mDZE5lnWLGXTP86hjEv+njPxEROUOzK5vx4KYHadCugdNRRMQBRflFpN2bRov4FvR6tJfTcQBo2qkpj2x7hMg6F58EotmVzRj8j8HsWLCDwtzCCm0jop4iEREB7OEpVpoFQMMODTHCNLmCSChyRbo4dfwUi55cxMGMg45myTuWx4p/rKCkqKRSxc01D1/DL2b+QgWRVJiKIhERAWDxHxfz0ciP2Ldmn9NRRMRBhmEw8q2R1IqpReotqRSdKnIkh9vtZvb9s/n8959z+LvDldrWMAwMw+D4ruOk3ZdGcUGxj1JKsFBRJCIi7FqyixXPr6Dbvd1o1aOV03FExGF1mtVh1LujOLTxEIt+t8iRDN9O/pZNn2xiwF8HVPn6xoPfHmTdxHV8/tTnXk4nwUZFkYhIiMs7lse0W6fR6NJGDH1pqNNxRMRPdLyhIz0e6sGG9zeQ+2NujbZ9bMcx5jw0h7Z929L7N72rvB9zpEm3cd1Y+Y+V7Fy804sJJdioKBIRCXGzH5jNyQMnSZ6arPH3InKGwf8YzP0b7qd2k9o11qbb7WbGXTMwXAZJk5MIc1Xv7erQfw6lccfGTL9tOnnH8ryUUoKNiiIRkRB3xU+vYPALgzVsTkTKiYiOIKZNDO4SN9ZMq0amuDYMgyEvDCFpchIxsTHV3l9knUiSpyZz8sBJlv55qRcSSjDSlNwiIiHKXeLGCDO44qdXOB1FRPzc5k838+lNn5L4ZiLd7unms3bys/OJqh9Fy+4tvbrflt1b8osZv6Bt37Ze3a8ED/UUiYiEoJKiEiYPnsya19c4HUVEAsAVP72C9gPbM+/ReRzZcsQnbeSfyOe/3f7Lkj8v8cn+O97Qkci6kRTmFXJi/wmftCGBS0WRiEgIWv7scnZ+vpOo+lFORxGRAGCEGYx+fzSuKBepY1MpLvT+FNfzHp3H8Z3H6ZDQwev7LuV2u5k8aDKfjPmEkqISn7UjgUdFkYhIiMn8KpOlf15K55s70yWli9NxRCRA1G9dn8SJifyw5gevX5uz+bPNrH93Pdf/7npir4/16r7LMgyDHg/1IPPLTJY/s9xn7UjgUVEkIhJCCk4WkJqSSv1W9Rnx2gin44hIgLnip1dw3f9dR/uB7b22z+zMbNLuTaNlj5b0e7qf1/Z7PnG3xBF3SxxL/7KUzFWZPm9PAoOKIhGRELJ9wXaydmeRNDmJWg1qOR1HRALQoGcHnS6KvDEb3aGNh3BFukiemowrwlXt/VXEDa/eQP3W9UlNSSX/RH6NtCn+TUWRiEgI6ZTciUe2PaIZmESk2hb/cTEz7phR7f1cOuxSHt35KI07NvZCqoqp1aAWSZOTqNOsDqeOn6qxdsV/qSgSEQkB2fuy2bVkFwAN2jZwNoyIBAXDZbBh0gYyPsyo0vYH1h9g3VvrcLvdRERHeDndxbXt05a7Vt5FTJvqfxeSBD4VRSIiQc5d4mbGHTP4MPFD8o7q29xFxDv6ju9L62tbM/uB2RzffbxS2xbmFfLZLZ+x+I+Lyc92bviaYRjkHc1jxt0zyN6X7VgOcZ6KIhGRILfqlVXsWLSDIS8OIbpRtNNxRCRIhIWHkTwlGXexm+m3TaekuOJTXC/87UJ+/O5HRr8/mloxzl7fmPtjLps+2sT026fjLqn+NVISmFQUiYgEsYPfHiT9/9IxR5l0u9d330IvIqGpYYeGDP/PcDJXZbJ/7f4KbbN17lbW/GcNPR/ryU8G/8THCS+u8WWNGfryUHam72TVy6ucjiMOCXc6gIiI+EbRqSJSU1KJbhRN4puJGIbhdCQRCUJdb+tK275tadi+4UXXLcgpYOZdM2kW14xBzw6qgXQV0+2ebmydvZX036XTPqE9l3S9xOlIUsPUUyQiEqRckS6uvvtqRr07ijpN6zgdR0SClGEYpwuiLbO2UJBTcN51I+tEkvhmIslTkwmv5T+fzRuGwci3RhLdKJp5j85zOo44wH8ejSIi4jXuEjdGmEGvx3o5HUVEQsTh7w7z4cgPiR8Xz41v3Fju9rxjeUQ3jOayGy9zIN3F1W5Sm5um36QZOkNUlXuKTNN8yJtBRETEO3J/zOX1uNfZOner01FEJIQ07dSU3k/0Zu1/12LNtM647cfvf+SVdq+w8eONDqWrmNY9W1P3krqUFJdwdNtRp+NIDarO8LmBpmkOLf3FNM26pmmmVmYHpmm+YJrme9XIICIiZbjdbmbeM5Oj245Sv1V9p+OISIgZ8NcBXHLVJcy8eyYnD5wEoLigmNSUVMIiwgLmi6PnPDSHd65/h5xDOU5HkRpSneFztwHzTdPMBEqAD4B/VnRj0zQTgNuB2dXIICIiQMbUDNLHp5O1OwuAzimdad6lucOpRCTUhEeFk/xBMhO7TWTy0MnkH88na4/9vNTrsV7Ua1HP4YQV0+PBHqx/dz1Thk0h70geWXuziImNIWFCAnEpcU7Hq5TTrw97AvcYakKle4pM0/yXaZp3ASZwNzAZmArcZlnW5AruoxEwAXimsu2LiMiZMqZmkDYu7XRBBGBNs8iYWrVvmRcRqY6mnZoSNzaOI98fOV0QAayduDZgnpead2lOp5914sA3B+xjcEPW7izSxqUFzDHAWa8PAXoMNaUqw+fmA82B3wL/D2gHHABGmKZZ/qq6c/svMB44VoX2RUSkjPTx6RTmFp6xrDC3kPTx6Q4lEpFQt2PhDooLis9YFmjPS3uX7y23LNCOQa8PFWe43Rf+5l7TNHsAhZZlrT/P7VFAZ6ArEGdZ1q8usr97gCssy/q1aZp3AP0ty7rjYkHXrl3bDth5sfVERELNrB6z4FxP5QbcuKain1WJiHhPMDwv6RiCQvv4+PhdFVmxItcUPQesAU4XRaZp3gqMBQ4Bz1mWtRZYW8FwNwEtTNNcDzQC6pqm+dLFiqlSnTt3JioqqoJN+dbatWuJj493Oob4mM5z6AjUc72k+ZLTFzSXFRMbE5DH42uBep6l8nSunbM8dvkZQ3pL+eJ5yVfnuSaPwVeWtlzKiX0nyi0PpGMoq6LnOj8/n40bKzfTYUWGz3UGZpT+YppmV+BdoD3QD1hhmma7ijZoWdZgy7I6W5Z1FfBHYGZFCyIRETnTqeOn7CEqxpnLI2pHkDAhwZlQIhLyEiYkEFE74oxlgfa8dK5jcEW5AuoYBv99MOHRZ/aBBNp5qCkVKYrqA/vK/D4W+B57ooUOwArg/7wfTURELmbOQ3M4lXWKfk/3I6ZtDBgQ0zaGxImJml1IRBwTlxJH4sTEgH5eOvsYwsLDiKgTQccRHZ2OdlHFBcWseW0NV950JSPfHPm/8xAbeOehplRk+NxeoBWwx/P7QOBTy7LcQJFpms8DE6vSuGVZ7wHvVWVbEZFQl/FBBhkfZND/L/3p91Q/+j/d3+lIIiKnxaXEBfyb77LHsG/1Pt7u/TazH5hN8gfJGIZxka2ds/jpxax4bgWNL2tc7jzkHMrhq39/Rc9HejqY0P9UpKdoAfAbANM0O2BPqLCwzO07gTbejyYiIueTtSeL2Q/Mpk3vNvT5XR+n44iIBL1W17Si/5/6s/GjjX49pfWupbtY8fcVXH3P1XQY1KHc7eveWse8X85j0yebHEjnvypSFD0DXGea5j7gK2A3sLLM7S2A8ldwiYiIz9RpXofuD3YnaUoSYeFV+XYFERGprOt/dz1trmvDgicWUHSqyOk45Zw6foppt06j0U8aMeylYedcp/dvetPqmlbMum8WWXvLTyQRqi76SmpZ1g9AD+AjYCaQ7Bk6VyoB2OKbeCIicraSohLCo8IZ9OwgGrZv6HQcEZGQEeYKI3lKMrcvvp3wWhW5CqVmzX1kLif3nyT5g2Qi60aecx1XhIvkqckUFxYz/bbplBSX1HBK/1Shs2lZ1h7g8fPc3An41GuJRETkvH74+gc+u+Uzbkq9iWadmzkdR0Qk5DRo1+D0/w9tOkSzK/3nubjnoz2J7RtLqx6tLrheo0sbMfxfw5l590xW/2c1vR7tVUMJ/Ve1S1zLsm71RhAREbmwgpwCUlNSKTpVRL1W9ZyOIyIS0ta8toa5v5zL3SvvptU1Fy5CfK0ov4jwqHBadm9Jy+4tK7TNVXdeRd6xPLqkdPFxusCggegiIgFi/q/nc2TrEZImJRHdMNrpOCIiIS3uljjqtaxHakoqBScLHMtRUlzC5EGTWfjbhRdfuQzDMOj9eG9qN6lNSVGJX14jVZNUFImIBIDvp3/PuonruO6319Gufzun44iIhLxaDWqRNDmJo9uPMu9X8xzL8cVzX7Dniz0079q8StsXFxTzXr/3mP/4fC8nCywqikREAsDGDzfSolsLBvxlgNNRRETEo12/dlz35HV889Y3fDftuxpvf9/qfSx5egmdb+5c5WFwrkgXrXq14uvXvmbLrNCdO01FkYhIABjz4RhS5qXginQ5HUVERMoY8OcBdBjcgTBXzb6tLjhpX2dav1V9Rrw2olr7SngmgeZdmjPjrhmcPHjSSwkDi4oiERE/9v2M7znxwwmMMIM6Tes4HUdERM7iinQxdv5YzJFmjbZ7YP0Bcg7lMHrSaGo1qFWtfYVHhZP8QTL52fnMvGsmbrf74hsFGRVFIiJ+6tDGQ3x606csenKR01FEROQCDMPAXeJm5YsrWf2f1TXSZuz1sTy2+zHa9Wvnlf01u7IZg/8xmKPbj5JzKMcr+wwkKopERPxQ0akiPrvlM2rF1GLIi0OcjiMiIhdjwJ5le1jw+AIOZhz0WTMnfjjB+vfW43a7q91DdLZrHr6G+9bdR93mdb2630CgokhExA+l/z6dQxmHGPXuKOo007A5ERF/ZxgGiW8lUqthLVJvSfXJFNfuEjfT75jO7Adnk52Z7fX9G4ZBRO0ICnIKWDZhGUX5oTNNt4oiERE/syN9B6teWkWPh3vQ8YaOTscREZEKqtO0DqPeHcWhjYdY9DvvD33+6l9fsWPhDoa+NJSYNjFe33+pPcv3sPgPi1n81GKfteFvVBSJiPiZFt1acO0T1zL4+cFORxERkUrqOLwjPR7uwep/r+bo9qNe2+/Bbw+y6MlFmCNN4sfFe22/53LpsEuJvy+elS+sZOfnO33alr9QUSQi4ifcbjfFhcVEN4xmyD+GEBEd4XQkERGpgsHPD+bOZXfS6CeNvLK/kqISUsemUqthLRLfSsQwDK/s90KGvDiExh0bM+22aeQdzfN5e05TUSQi4ie+eecb3u71NjmHQ2/WHxGRYBIRHUGb3m0Ae+rs6k5xHRYeRsKzCSRNTqqxr2eIrBNJ8gfJ5BzMYf6v5tdIm05SUSQi4geObD3CvEfnUatBLWo3ru10HBER8YLdy3bz327/5Zu3v6nyPgrzCgG4bMRl/GTwT7wVrUJaxrdk1Huj6PvHvjXarhNUFImIOKy4sJhpY6fhinQx+v3RGGG+HxYhIiK+F3t9LO0Htmfeo/M4suVIpbfP/TGXVy9/lXVvr/NBuorpktKFRj9phNvtJj8737EcvqaiSETEYcv+uox9q/eRODGR+q3rOx1HRES8xAgzGP3+aFxRLlLHplJcWFzhbd1uNzPvmcnJAydp2b2lD1NWzMy7ZjJl2BRKikqcjuITKopERBxUXFCMNcPiqjuu4oqfXuF0HBER8bL6reqT+GYiP6z5gaV/Xlrh7da9tQ5rhkXCswlc0vUSHyasmA5DOpD5ZSbLJixzOopPhDsdQEQklLkiXdzz1T1B+8mbiIjAFWOuoPsD3al7Sd0KrX9kyxHmPzafDoM60OuxXj5OVzFxN8exdfZWlv11GZcOvZTWvVo7Hcmr1FMkIuKQdW+tI/9EPuG1womsG+l0HBER8aERr43gmoevqdC6e1fuJbJuJKPeG+VX15ne8OoN1G9dn9SUVPJPBNf1RSqKREQcsPGjjaTdm8a6N527eFZERGrepk82MefhORdc56o7ruKRbY9Qv5V/XWdaK6YWyVOSKcwr5OhW730xrT9QUSQiUsOy9mQx6/5ZtL62NT1/2dPpOCIiUoN+/P5H1ry6howPM8rdtmfFHrbN2wZAVL2omo5WIbHXx/Lojkdp0a2F01G8SkWRiEgNKikuYdpt03AXu0mekkxYuJ6GRURCSZ/f96FN7zbMfmA2x3cfP738VNYpUlNSmfvIXIoLKj5LnRPCa4VTUlzCF3//guzMbKfjeIVejUVEatCql1axe+luhv97OA07NHQ6joiI1LCw8DCSJifhLnEz/bbplBTbE+3MeWgO2ZnZJE1JwhXpcjjlxWXtyWLZX5Yx/Y7puEvcTsepNs0+JyLiYxlTM0gfn07WnizqtaxHpzGd6Hp7V6djiYiIQxp2aMgN/7mB6bdP58UWL5J7OBeATmM60bpnYMzq1rB9Q4a9Moy0e9N4vvHznMo6RUxsDAkTEohLiXM6XqWpp0hExIcypmaQNi6NrN1Z4IYT+06wbe42Nn6w0eloIiLiIMNl4IpynS6IALbN3UbG1PLXGvmr8OhwDJfBqeOnwA1Zu7NIG5cWUMdQSkWRiIgPpY9PpzC38IxlhbmFpI9PdyiRiIj4g8/Hf05x/pnXDgXa68Pn4z/HXXzm0LlAO4ZSKopERHwoa09WpZaLiEhoCIbXh2A4hlIqikREfKheq3rnXB4TG1PDSURExJ+c73UgkF4fguEYSqkoEhHxEbfbTXSj6HLLI2pHkDAhwYFEIiLiLxImJBBRO+KMZYH2+hAMx1BKRZGIiI9k780m93AuXW7vQkzbGDAgpm0MiRMTA3JmHhER8Z64lDgSJyYG9OtDMBxDKU3JLSLiIzGxMTy0+SGiYqIwDMPpOCIi4mfiUuICsoAoKxiOAdRTJCLidUX5RXz1r68oLiymVoNaKohERET8nIoiEREv+3z858x7dB57vtjjdBQRERGpABVFIiJetCN9B1+++CXdH+hO+wHtnY4jIiIiFaCiSETES/KO5jH99uk0ubwJQ14Y4nQcERERqSAVRSIiXjLn4TnkHMoheWpyuSlKRURExH9p9g1gFyEAACAASURBVDkRES/p/URvOgzuQItuLZyOIiIiIpWgokhEpJoK8wqJiI6gRbcWKohEREQCkIbPiYhUQ3FhMZMGTmLBEwucjiIiIiJVpKJIRKQalv1tGZmrMmnZo6XTUURERKSKVBSJiFTR3pV7Wf635XS5tQudb+rsdBwRERGpIhVFIiJVkJ+dT+rYVGJiY7jhPzc4HUdERESqQUWRiEgVHNhwgFPHTpE0JYmo+lFOxxEREZFq0OxzIiJV0LZPWx7b/ZgKIhERkf/f3p0HRlWefR//neyEQADDLqACHrYIsloWFcISlBAIrQtxwQ0f9/q6tBb7VltRW9eqVAsoKobSpyXsFJAEAQERo0AQOCD7vhMgIft5/ghQkCWTZGbOLN/PPzonM/f8bs8cM1fue64JAKwUAUAF5OzMUda4LNm2TUEEAECAoCgCABfZpbam3TtN856epxO7TzgdBwAAuAnb5wDARcveWqZtC7cpaXySal5Z0+k4AADATVgpAgAX7P1hrzJHZap1Smtdf//1TscBAABuRFEEAOUoLS5Vemq6qtetrkFjB8kwDKcjAQAAN2L7HACUIyQsRP3f6q+wqDBFXxHtdBwAAOBmFEUAcBmFuYWKqB6hlgNbOh0FAAB4CNvnAOASTu4/qQ+u/UDfj//e6SgAAMCDKIoA4CJs29aM+2co73CervzFlU7HAQAAHsT2OQC4iO8+/E6b5mxS4l8TVa9tPafjAAAAD2KlCAB+5uD6g5r/zHw1H9BcXZ/o6nQcAADgYRRFAPAzu1fsVlTtKCVPSKb9NgAAQYDtcwDwMx1GdFCbX7VRRPUIp6MAAAAvcKQoMk3zD5JuO31ztmVZzzuRA/4tOy1bGaMylLMjR7FNY5UwOkHxqfFOx6oQ5uAbzs5he44y62Yq8Z1Ev5sDAACoPK9vnzNNs6+k/pKul9RBUifTNId6Owf8W3ZatmaOnKmc7TmSLeVsz9HMkTOVnZbtdDSXMQffcN4cJOUdzPO7OQAAgKpx4jNFeyU9Y1lWoWVZRZLWS2rqQA74sYxRGSrKKzrvWFFekTJGZTiUqOKYg28IhDkAAICqMWzbduzJTdNsKWmppB6WZW263H2zsrKukrTVG7ng+2Z1mSVd7KVrSINWDvJ6nspgDr4hEOYAAAAu6upOnTptc+WOjjVaME2zraTZkp4rryA6V7t27RQZGem5YBWQlZWlTp06OR0jKC1uvFjHdx2/4Hhs01i3nxNPnWdvzsFTFjVapBO7T1xwnDnAl/H/7uDBuQ4OnOfg4eq5Ligo0Nq1ays0tiMtuU3T7CEpQ9JvLcv6zIkM8G99Xu2jkPDzX77h0eFKGJ3gUKKKsUttRdSMkBFyfrvn0MhQv5lDaUmpImIu7M7mT3OQpL5/7qvQiNDzjvnTawkAAFSdE40WmkiaJmm4ZVmTvf388H8lhSVqf3d7DZkwRLHNYiWj7K/6SWOT/KZj2DfvfqND6w6p/X3tz84hNCJUIWEhanZzM6fjuWTZG8t02DqsjiM7np1DSHiIQsJDdHXC1U7Hc0nB8QJdl3qdkj9J/u9rqZl/vZYAAEDVObF97llJUZLeNk3zzLGPLMv6yIEs8DMHfjygtMQ0DZs8TPGp8ee9cT288bDmPDFHie8mKiTUd7+XeN/qfcp4IUOthrTS4HGDz3456OGNhzW281htWbBFHe7t4HDK8jXu1lhdn+yqxHcTz87h4PqDGttxrKbfN13D5wz36S8+3fXNLqUNTNPtU28/+1piCwYAAMHJ60WRZVlPSXrK288L/1dcUKz04ekqLihWnRZ1Lvj5zuU7tfKDlarZuKZ6/ranAwnLV3SqSOnD01XtimpKGpd0XtFwxbVX6KktTyk6LtrBhOWzbVuGYejq3lfr6t7nrwjVbV1X/d7sp/88/h+tHLNSXR/v6lDKyys4UaD01HRFxkaqQYcGTscBAAAO890/pwM/kzkqU/vX7FfyJ8mKqR9zwc/b39NebX7VRgt/v1B7svY4kLB8+UfzFREToSGfDrlo8XPm2NbMrdq/Zr+347lkzuNzlDEqQ5fqXNnl0S5qeUtLLfrjIhXmFno5nWvmPjVXx7Yd09CJQxVVK8rpOAAAwGEURfALWxZs0fK3lqvzo5117aBrL3ofwzA06KNBql6/utJT0y/47hlfUKNRDT2w/AE179/8kvcpzi/W1HumasqdU1R0yrfmYM2w9N3fvlNJYcklt8YZhqHBnwzWA8sfUET1CxsxOG3dv9dp1YRV6vHbHmrWyz8+vwUAADyLogh+wZphKa5VnPq/0f+y96tWp5qGThyqwxsPa+kbS72Urny5B3M1c+RMnTpy6oKOcz8XFhWm5E+SdXDdQS34zQIvJSzfyX0nNeOBGWrQoYH6vNLnsveNqR+jOs3ryLZt7Vqxy0sJXbMna48adWmkm1+62ekoAADAR1AUwS8k/jVR9y+9X+HR4eXe9+reV+v2qberx/M9vJCsfLZta8YDM7T689U6vvvC7yW6mOb9m6vbU9307fvf6qe5P3k4Yfls29b0+6ar8GShUialKCzStY8jZo3N0sc3fKwtGVs8nNB1fV/rqxGLRig0PLT8OwMAgKBAUQSfZs2wdMg6JMMwVK1ONZcf1yq5lcKrhavwZKHyDuV5MGH5ssZmaePMjer7el/Vj6/v8uP6vt5X9drV07QR03TqyCkPJizf/tX7tTVzq/q92U91W9d1+XHt726vuFZxmnav83NY/fnqs581C69WfnENAACCB0URfNaRn45oyvApmv/M/Eo9vrSkVB93/1hT7556yaYAnnZowyHNe3pe2crPk90q9NiwqDClTEpRr9/1crwZQIMODfToj4+qy6NdKvS48OhwpaSlKPdArmY9PMux87Bv9T7NfGimlr7uO1sqAQCA76Aogk8qKSpR+l3pCg0P1a0f3lqpMUJCQ9T5fzrrp7k/aeWYlW5O6Jp5T89TeHS4kj9NLvezRBdTP76+uj3ZTUaIoZLCEg8kvLzigmJtmrNJklSnRZ1Kfe9Qw44N1eeVPlr373Va/dlqd0cs19k26HWqVfq1BAAAAhtFEXzS4lcWa/eK3Rr090GKbRJb6XE6P9JZLW9tqS+f+1IHfjzgxoSuSZ6QrDum3aEaDWtUaZytC7fqr9f8VYesQ25K5pqM32Vo0q2TtG/VviqN0/3Z7mp3RzvFNLiwlbqnLfjNAh1cd1DJnyb7/HdAAQAAZ1AUwefs/na3lryyRO3vaa+2t7Wt0liGYWjwx4MVUSPi7Be/esOxbcdUWlKqmAYxatqzaZXHizPjVJxfrPTUdK+tGG1ZsEXfvP2NOj/SucpfcGqEGBr2j2FqkdjCTelcs33Jdn37/rfq9lQ3tRjg3ecGAAD+g6IIPqdefD31erGXBr4/0C3jxdSPUfKEZEXWjFT+sXy3jHk5+cfyNeHGCZr50Ey3jVmjUQ0ljUvS3qy9+uqlr9w27qXkHc7TtHunlbVBf/PybdArwrZtLX5lsRb9aZHbxrycJr9oosT3EtX39b5eeT4AAOCfKIrgU4rzixVeLVy9X+6tyJqRbhv32luv1YjFIxRT37Pbt2zb1uxHZuvEnhPq/D+d3Tp266Gtdf2D1+vr17/W9sXb3Tr2uWzb1qyHZyn3YK5S0lJcaoPuKsMwdGTTES16aZF2LtvptnF/zrZt5R/LV0hYiLo90U1hUa61EAcAAMGJogg+48f//VF/a/c3Hdt2zCPjG4ah3AO5mn7/dOUd9kyb7uy0bK2dvFY3v3SzGndt7PbxE99JVJ3mdc42P/AEwzDUOqW1Brw9QA07NnT7+APfH6jYZrFKvytdBccL3D6+JH0/7nuNaT1GR7ce9cj4AAAgsFAUwSfk7MzRrIdnKTouWjWvrOmx5zm+67jWfLHGI+2hj249qjmPzVHTnk3V84Webh37jIiYCD347YMe2w525r9J/PB4dX28q0eeI7JmpIZOHKqc7Tn6z5P/cfv4h6xDmvvruaoXX0+1mtVy+/gAACDwUBTBcaUlpZp2zzSVFpcqJS1FIWGee1k27NhQvf/UW+unrNeqT1e5dez8Y/mq3by2hk4cqpBQz82hWu2yL7E9sPaANs7a6LZxS4pK9Hmfz/XDJz+4bcxLadqjqXqN6qU1X6zRoQ3u66hXUlii9NR0hUeHa8inQyrVBh0AAAQfiiI4bvlby7Xtq21KfK9sa5indX+2u5rd1Exzn5yrI5uPuG3chtc31Miskap1lXdWJ+Y9PU/pqelu2264+E+Lte2rbYqIiXDLeOW58fc3auR3IxXXKs5tY3710lfam7VXSeOSVKNR1dqgAwCA4EFRBEfZpbY2z9us1sNaq8OIDl55zpDQEA39fKiMUEPzn5lf5fF2f7tbc5+eq+L84kp9uWllJY1LkiRNvXuqSktKqzTWjqU7tGS0e9qguyo0PPRsq+8dS3fILq3adka71Nah9Yd0/YPXq/XQ1u6ICAAAggRFERxlhBi6a/5dZVudvFhQxDaN1Z0z7tTg8YOrNE7hyUKlp6Zr/ZT1KjpV5KZ0rql1VS3dMuYW7fh6h75+/etKj1NwvEBT75qq2GaxbmuDXhE7l+/UhJ4TtOytZVUaxwgxdFv6bbp1zK1uSgYAAIIFRREckzUuSyf3n1RIaIjXtmydq9mNzRQdF63S4tJKdymb+3TZFryhE4ee/ayPN8WnxqvdHe206KVF2vvD3kqNsWnOJh3fdVxDJw51axt0V115w5VqndJamaMyKz2HJa8u0dGtR2UYhkIjQt2cEAAABDqKIjhi46yNmjVyllaOWel0FKXfla7PEz6vcHvo9VPX64fxP6jHb3roqpuu8ky4chiGoVs/vFU3/v8bVbdN3UqN0e6Odnp84+Nq2qOpm9O5xjAMDRo7SNXrVlf68HQV5VVsxW1N2hpljsrU2slrPZQQAAAEOooieN3J/Sc1/f7pqt++vnqN6uV0HHV9omtZe+gnXG8PXVJYorlPzS3rZvdybw+mK19UrSjd9PubFBYZVqEtfDk7c7Rj6Q5JUu2ra3sqnkuir4jWkM+G6NCGQ5r/nOuf8zq27ZjmPFrWBr3H8z08mBAAAAQyiiJ4lW3bmnH/DBWeKFRKWorCIsOcjnS2PfTqz1frx//90aXHhEaE6u75d2vY5GE+s13ryE9HNKbVGK2fur7c+55pgz7p1knKz8n3QrryXdP3GvV6sdfZ5gvlKS0p1dS7p0qSx9ugAwCAwMa7CHjVDx//oE1zNqnfG/1Ur209p+OcdePvb1Tjro016+FZytmZc9n7HrIOybZtxbWK0xUtr/BSwvLFNo1VdFy0Zj40Uyf2nLjsfc+0QR/w9gBFxUZ5KWH5+vypjzo91Mml+67820rt+HqHbhlzi9faoAMAgMBEUQSvajWklfq82kddHuvidJTzhIaHKiUtRbWb11ZBzqU/W7R/zX59dN1H+ubdb7yYzjWhEWVzKMor0vT7pl+yxfXe7/cq88XMsjbo93mnDXpFrZ64WlPunCLbvnSb7uvvu15J45IUnxrvxWQAACAQURTBK0oKS1RaXKrouGj1eqGXV9tvu6pOizp6aOVDqtfu4itYxfnFSk9NV1TtKF1313VeTueauFZxGvDOAG2ev1kr3ltxwc+L8oqUnpqu6nWra9DfB/nkeZCk/GP5Wjt5rb778LsLflaYW6iivCJFxESo44MdfXYOAADAf1AUwSsyX8zUhF4TVJxf7HSUyzIMQ4W5hZr92Gzt/f789tALfrtAB9YeUPKEZFWvW92hhOXrNLKTzMGmtizYcsFKS2hkqDrc10FDPhui6CuiHUpYvq6Pd1WLxBaa/8x8HVx/8Lyfzf31XI3vNt7nX0sAAMB/UBTB47Yu3Kplby5T/Q71FRblfGOF8hTnF8uaZik99b/toTfP36wVf12hLo93UcuBLR1OeHmGYSglLUV3zrjzvFUUu9RWSGiIejzfQ9f0vcbBhOUzDEPJE5IVEROh9NR0lRSWSPpvG/SWg1r6xWsJAAD4B95VwCOy07KVMSpDOTtyZIQYql6/uga8NcDpWC450x56Yr+JerP+myrMLVR0XLTqmHXU7y/9nI7nkjNfhrvigxVa9IdFOnX0lEJCQ/SL536hvq/2dTida2IaxGjwx4M1OXmy3m78tvIO58kwDMU2i3W8DToAAAgsrBTB7bLTsjVz5EzlbM+RbMkusVVwrEAbpm5wOprLcvfnKiQsRIUnCyVbyjuYpxM7T2hDuv/MITstW/Ofnq9TR05JtlRaXKoV76xQdlq209FcVniiUGFRYco7lFf2Wiq1lbs/V+v+tc7paAAAIIBQFMHtMkZlnN12dkZxfrEyRmU4lKjiMkZlqLS49LxjRXlFfj8HfzwPP//skL/NAQAA+D6KIrhdzo6Lf8/PpY77IubgGwJhDgAAwPdRFMHtYpvEXvx404sf90WXysocvCsQ5gAAAHwfRRHcrmHnhhccC48OV8LoBAfSVE7C6ASFR4efd4w5eF8gzAEAAPg+iiK41a5vdsmabqlJjyaKbRYrGVJss1gljU1SfGq80/FcFp8ar6SxSczBYYEwBwAA4PtoyQ23KThRoPTUdNW8sqaGzx6uqNgopyNVSXxqvOJT45WVlaVOnTo5HadSzszBnwXCHAAAgG+jKILbfPn8lzq27ZhGLBrh9wURAAAAggdFEdym+7Pd1ahzIzXt2dTpKAAAAIDLKIpQZYUnCxVePVx1mtdRneZ1nI4DAAAAVAiNFlAldqmtyUMma+rdU52OAgAAAFQKRRGqZPk7y7U1Y6ua3dTM6SgAAABApVAUodL2rdqnjBcy1GpIK3V8sKPTcQAAAIBKoShCpRSdKtKU4VMUHRetpHFJMgzD6UgAAABApdBoAZVyZNMR5R/N15DPhig6LtrpOAAAAEClURShUupfV19Pbn5S4dHhTkcBAAAAqoTtc6iQ3AO5WvrGUpUWl1IQAQAAICBQFMFltm1rxgMztPD3C3Vk8xGn4wAAAABuwfY5uCzr71naOGujBrwzQHFmnNNxAAAAALdgpQguObThkOb9v3lq3r+5uj3Zzek4AAAAgNtQFKFctm1r2ohpCo8OV/KnyTJCaL8NAACAwMH2OZTLMAwNfG+gTh09pRoNazgdBwAAAHAriiJcVn5OvqJio9S4a2OnowAAAAAewfY5XNKpo6f00XUfaclrS5yOAgAAAHgMRREuyrZtzX5ktk7sOaFr+l7jdBwAAADAYyiKcFFrvlijH//5o2566SY17sLWOQAAAAQuiiJc4OjWo5rz2Bw17dlUPX/b0+k4AAAAgEdRFOECB9YeUGSNSA2dOFQhobxEAAAAENjoPocLmEmmmvdrrrAoXh4AAAAIfCwD4Kzd3+7Wqk9XybZtCiIAAAAEDd75QpJUeLJQ6anpKiksUZtftlFETITTkQAAAACvYKUIkqS5T8/Vkc1HNOTzIRREAAAACCoURdD6qev1w/gf1OM3PXTVTVc5HQcAAADwKke2z5mmOVzSi5LCJb1rWdYYJ3JUVnZatjJGZShnR46WNF2ihNEJik+NdzpWhZw7B8MwFHtVrHq/3NvpWAAAAIDXeX2lyDTNxpJGS+opqYOkkaZptvF2jsrKTsvWzJEzlbM9R7KlnO05mjlyprLTsp2O5rKfz8EutZW7L1fr/rXO6WgAAACA1zmxfa6vpEzLso5YlpUr6d+SfulAjkrJGJWhoryi844V5RUpY1SGQ4kq7mJzKM4v9qs5AAAAAO7ixPa5RpL2nnN7r6Surj74hRde0MmTJ90eylUbt2+8+A+2S+tG+sdKSyDMAQAAALiYmJgYpaamVugxThRFIZLsc24bkkpdffBrr72myMhIt4dy1bvz3y3bdvYzsc1i9euxv3YgUcW9M/cdHd95/ILj/jQHb8rKylKnTp2cjgEv4FwHB85z8OBcBwfOc/Bw9VwXFBRo7dq1FRrbie1zuyQ1POd2A0l7HMhRKQmjExQeHX7esdDIUCWMTnAoUcXVbVP3gmPh0eF+NQcAAADAXZwoihZISjBNs65pmtGShkma60COSolPjVfS2CTFNouVDCkkLERh1cLUPLG509Fcsnn+Zm2et1nX9L/m7Bxim8UqaWyS33XQAwAAANzB69vnLMvabZrmKEkLJUVIGm9Z1rfezlEV8anxik+NV1ZWlhoZjTT+hvFaO3mtuj7m8kejHHNw3UHVb19fd0y7Q+HVwst/AAAAABDgHPmeIsuyJkma5MRzu1vDjg31SPYjijPjnI7ikht+fYO6PNZFoeGhTkcBAAAAfIIT2+cCzpmC6MDaAzqy+YjDaS4ue1K2Ns/fLEkURAAAAMA5KIrcpLigWBP7T1R6arpKikqcjnOewxsPa+ZDM7XszWWybbv8BwAAAABBhKLITcIiw5T4bqJ2r9itxa8sdjrOWSVFJUpPTVdYVJiSJyTLMAynIwEAAAA+haLIjdre1lbt72mvJa8s0c5lO52OI0n66qWvtOe7PRo0dpBqNq7pdBwAAADA51AUudnA9wcqtlms0u9KV8HxAkez7F+zX1+/9rU63N9BbYa1cTQLAAAA4Ksc6T4XyCJrRirlixRtmLZBoZHONjSoF19PyZ8kq80vKYgAAACAS6Eo8oAm3ZuoSfcmkiTbth35HE/uwVxVr1tdHUZ08PpzAwAAAP6E7XMetHvlbo3tOFY5O3O8+rzZk7L1XvP3tH/Nfq8+LwAAAOCPKIo8qFrtajq86bCm3TtNdql3WmEf235Msx+Zrfrx9VW3TV2vPCcAAADgzyiKPKhOizoa+N5AbVu4TcveWubx5ystKdXUu6fKtm0N/WKoQsI4vQAAAEB5eNfsYR3u66DWw1orc1Sm9n6/16PPtfQvS7VjyQ7d8sEtqn11bY8+FwAAABAoKIo8zDAMDfr7IFWvW11ZY7M89jy2bStnR47a3t5W1919nceeBwAAAAg0dJ/zgugronX/0vsV2zTWY89hGIYGfThIpcWljnS7AwAAAPwVK0VeUuuqWjJCDB3ffVw7l+1069hLXl2ifav2SRKfIwIAAAAqiHfQXjbtnmn659B/KvdArlvG2zB9gzJHZWrtP9e6ZTwAAAAg2FAUeVnie4kqOF6g6fdPl21XrU33ib0nNOOBGWrYsaF6v9zbTQkBAACA4EJR5GX12tZT37/01abZm/Tdh99Vehy71Nb0+6arKK9IKWkpCo0IdWNKAAAAIHhQFDmg6+Nd1SKxheY/M18H1x+s1BjZk7K1ed5m9X+rv+Jaxbk5IQAAABA86D7nAMMwlDwhWV+99JVqNKxRqTHa3t5WpSWlan9PezenAwAAAIILRZFDYhrEaNBHgySVbYUzQlxro11cUKyi3CJVq1NNHe7t4MmIAAAAQFBg+5zDcnbmaHy38dq6cKtL98/4XYY+av+R8o/lezgZAAAAEBwoihxWrU41FRwv0LR7punU0VOXve/mLzfrm7e/kZlsKqpWlJcSAgAAAIGNoshhEdUjlDIpRSf3ndSsh2ddsk133uE8TR8xXXGt49TvjX5eTgkAAAAELooiH9CoUyP1/lNvrfvXOq3+fPUFP7dtWzMfmqncg7kaNmmYwquFO5ASAAAACEwURT6i+3Pd1ezGZvp+7PcXrBYVnypWUV6REl5NUIMODRxKCAAAAAQmus/5iJDQEP3qX79SZM1IGcb5nejCo8OVOifVoWQAAABAYKMo8iHV61WXJH0//ntl/C5DeYfyFBEToT6v9FG3J7s5nA4AAAAITGyf8zHZadma/chs5R3Mk2yp8EShvnzuS2WnZTsdDQAAAAhIFEU+JmNUhkqLS887VlJYooxRGQ4lAgAAAAIbRZGPydmRU6HjAAAAAKqGosjHxDaNrdBxAAAAAFVDUeRjEkYnKDz6/O8hCo8OV8LoBIcSAQAAAIGN7nM+Jj41XlLZZ4tyduQotmmsEkYnnD0OAAAAwL0oinxQfGo8RRAAAADgJWyfAwAAABDUKIoAAAAABDWKIgAAAABBjaIIAAAAQFCjKAIAAAAQ1CiKAAAAAAQ1iiIAAAAAQY2iCAAAAEBQoygCAAAAENQoigAAAAAENYoiAAAAAEEtzOkAFRAqSYWFhU7nOE9BQYHTEeAFnOfgwbkODpzn4MG5Dg6c5+Dhyrk+p14IdXVcw7btSkbyrqysrJ6SljidAwAAAIBf6NWpU6evXbmjP60UrZTUS9JeSSUOZwEAAADgm0IlNVRZ/eASv1kpAgAAAABPoNECAAAAgKBGUQQAAAAgqFEUAQAAAAhqFEUAAAAAghpFEQAAAICgRlEEAAAAIKhRFAEAAAAIav705a0+wzTN4ZJelBQu6V3LssY4HAkeYprmQkn1JBWdPvSwZVkrHIwENzJNs6akZZIGWZa1zTTNvpLellRN0j8ty3rR0YBwi4uc5wmSekrKPX2Xly3LmupYQLiFaZp/kHTb6ZuzLct6nms68FziPHNNByDTNP8o6ZeSbEkfW5b1tievab68tYJM02ws6WtJnSQVqOwX7Z2WZa1zNBjczjRNQ9IuSc0syyp2Og/cyzTNbpLGSWol6VpJ+yVZkm6StFPSbJX90eM/joVElf38PJ8uirIl9bcsa6+z6eAup98ovSypt8reQM2VNF7Sn8U1HTAucZ4/kPRHcU0HFNM0b5I0WtLNKluEWCdpiKSZ8tA1zfa5iusrKdOyrCOWZeVK+rfKqlgEHvP0P+ebprnaNM3HHU0Dd3tI0mOS9py+3VXSJsuytp4ugr+Q9CunwsFtzjvPpmlGS2oq6RPTNNeYpvmyaZr8LvR/eyU9Y1lWoWVZRZLWq+yPHVzTgeVi57mpuKYDjmVZiyT1Pn3t1lPZ7rZa8uA1zfa5imuksovyjL0qezOFwFNbUoakJ1T2V4qvTNO0LMv60tlYcAfLsh6UJNM8U/te9Nq+0sux4GYXOc8NJGVKelRSjqRZkh5Q3eDlAAAAAzVJREFU2WoS/JRlWT+e+XfTNFuqbHvV++KaDiiXOM+9VLaawDUdYCzLKjJN82VJz0r6lzz8e5qiqOJCVLZke4YhqdShLPAgy7KWS1p+5rZpmh9LukUSRVFg4toOApZlbZE09Mxt0zTfl3SPeAMVEEzTbKuyLTXPSSpW2WrRGVzTAeLc82xZliWu6YBlWdYfTNP8s8q2zV0rD/6eZnmx4nZJanjO7Qb67/YbBBDTNHuapplwziFD/224gMDDtR0ETNOMN01z2DmHuK4DhGmaPVS2uv9by7I+E9d0QPr5eeaaDkymabYyTbODJFmWlScpXWUrgh67plkpqrgFkl4yTbOuyrqcDJM00tlI8JBakv5ommZ3lW2fu1fS/zgbCR60QpJpmmYLSVslDZf0ibOR4AGGpHdN08yUdFJl///+zNlIqCrTNJtImibpdsuyMk8f5poOMJc4z1zTgekaSS+bptlTZatDyZL+LukNT13TrBRVkGVZuyWNkrRQ0ipJkyzL+tbZVPAEy7JmqWx5/gdJWZI+Ob2lDgHIsqx8SSMkTVFZl5sNKmukggBiWdYaSa9JWqqy87zKsqx/OJsKbvCspChJb5umuco0zVUqu55HiGs6kFzsPHcX13TAsSxrjs5/D7bMsqzJ8uA1TUtuAAAAAEGNlSIAAAAAQY2iCAAAAEBQoygCAAAAENQoigAAAAAENYoiAAAAAEGN7ykCAPgd0zSvUtn3VLS0LOsnh+MAAPwcK0UAAAAAghpFEQAAAICgRlEEAPB7pmmONE3zpGmaNzidBQDgf/hMEQDAr5mmOVjSO5KSLcv6xuk8AAD/w0oRAMCf/UJSmqR7LMta4HQYAIB/YqUIAODPxqnsd9l2p4MAAPwXK0UAAH/2B0n/kPQ30zT5nQYAqBR+gQAA/NkUSb+R1FrSQw5nAQD4KYoiAIBfsyxrj6TRkl41TTPO6TwAAP9DUQQACATvSDoi6S9OBwEA+B/Dtm2nMwAAAACAY1gpAgAAABDUKIoAAAAABDWKIgAAAABBjaIIAAAAQFCjKAIAAAAQ1CiKAAAAAAQ1iiIAAAAAQY2iCAAAAEBQoygCAAAAENT+D7EtUcHTaObgAAAAAElFTkSuQmCC\n", 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