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<title>Homework 1</title>
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<body>
<header id="title-block-header" class="quarto-title-block default page-columns page-full">
<div class="quarto-title-banner page-columns page-full">
<div class="quarto-title column-body">
<h1 class="title">Homework 1</h1>
</div>
</div>
<div class="quarto-title-meta">
<div>
<div class="quarto-title-meta-heading">Author</div>
<div class="quarto-title-meta-contents">
<p><span style="background-color: yellow;">Insert your name here</span> </p>
</div>
</div>
</div>
</header><div id="quarto-content" class="page-columns page-rows-contents page-layout-article">
<div id="quarto-margin-sidebar" class="sidebar margin-sidebar">
<nav id="TOC" role="doc-toc" class="toc-active">
<h2 id="toc-title">Table of contents</h2>
<ul>
<li><a href="#question-1" id="toc-question-1" class="nav-link active" data-scroll-target="#question-1">Question 1</a></li>
<li><a href="#question-2" id="toc-question-2" class="nav-link" data-scroll-target="#question-2">Question 2</a></li>
<li><a href="#question-3" id="toc-question-3" class="nav-link" data-scroll-target="#question-3">Question 3</a></li>
<li><a href="#appendix" id="toc-appendix" class="nav-link" data-scroll-target="#appendix">Appendix</a></li>
</ul>
</nav>
</div>
<main class="content quarto-banner-title-block" id="quarto-document-content">
<p><a href="https://github.com/psu-stat380/hw-1">Link to the Github repository</a></p>
<hr>
<div class="callout-important callout callout-style-default callout-captioned">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<div class="callout-caption-container flex-fill">
Due: Sun, Jan 29, 2023 @ 11:59pm
</div>
</div>
<div class="callout-body-container callout-body">
<p>Please read the instructions carefully before submitting your assignment.</p>
<ol type="1">
<li><p>This assignment requires you to:</p>
<ul>
<li>Upload your Quarto markdown files to a <code>git</code> repository</li>
<li>Upload a <code>PDF</code> file on Canvas</li>
</ul></li>
<li><p>Don’t collapse any code cells before submitting.</p></li>
<li><p>Remember to make sure all your code output is rendered properly before uploading your submission.</p></li>
</ol>
<p>⚠️ Please add your name to the the author information in the frontmatter before submitting your assignment.</p>
</div>
</div>
<p><br><br><br><br></p>
<hr>
<section id="question-1" class="level2">
<h2 class="anchored" data-anchor-id="question-1">Question 1</h2>
<div class="callout-tip callout callout-style-default callout-captioned">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<div class="callout-caption-container flex-fill">
20 points
</div>
</div>
<div class="callout-body-container callout-body">
</div>
</div>
<p>In this question, we will walk through the process of <em>forking</em> a <code>git</code> repository and submitting a <em>pull request</em>.</p>
<ol type="1">
<li>Navigate to the Github repository <a href="https://github.com/psu-stat380/hw-1">here</a> and fork it by clicking on the icon in the top right</li>
</ol>
<p><img src="images/fork.png" class="img-fluid"></p>
<blockquote class="blockquote">
<p>Provide a sensible name for your forked repository when prompted.</p>
</blockquote>
<ol start="2" type="1">
<li><p>Clone your Github repository on your local machine</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode bash code-with-copy"><code class="sourceCode bash"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="ex">$</span> git clone <span class="op"><<insert</span> <span class="ex">your</span> repository url here<span class="op">>></span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="st">$ cd hw-1</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<p>Alternatively, you can use Github codespaces to get started from your repository directly.</p></li>
<li><p>In order to activate the <code>R</code> environment for the homework, make sure you have <code>renv</code> installed beforehand. To activate the <code>renv</code> environment for this assignment, open an instance of the <code>R</code> console from within the directory and type</p>
<div class="sourceCode" id="cb2"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>renv<span class="sc">::</span><span class="fu">activate</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<p>Follow the instrutions in order to make sure that <code>renv</code> is configured correctly.</p></li>
<li><p>Work on the <em>reminaing part</em> of this assignment as a <code>.qmd</code> file.</p>
<ul>
<li>Create a <code>PDF</code> and <code>HTML</code> file for your output by modifying the YAML frontmatter for the Quarto <code>.qmd</code> document</li>
</ul></li>
<li><p>When you’re done working on your assignment, push the changes to your github repository.</p></li>
<li><p>Navigate to the original Github repository <a href="https://github.com/psu-stat380/hw-1">here</a> and submit a pull request linking to your repository.</p>
<p>Remember to <strong>include your name</strong> in the pull request information!</p></li>
</ol>
<p>If you’re stuck at any step along the way, you can refer to the <a href="https://docs.github.com/en/pull-requests/collaborating-with-pull-requests/proposing-changes-to-your-work-with-pull-requests/creating-a-pull-request-from-a-fork">official Github docs here</a></p>
<p><br><br><br><br> <br><br><br><br></p>
<hr>
</section>
<section id="question-2" class="level2">
<h2 class="anchored" data-anchor-id="question-2">Question 2</h2>
<div class="callout-tip callout callout-style-default callout-captioned">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<div class="callout-caption-container flex-fill">
30 points
</div>
</div>
<div class="callout-body-container callout-body">
</div>
</div>
<p>Consider the following vector</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>my_vec <span class="ot"><-</span> <span class="fu">c</span>(</span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a> <span class="st">"+0.07"</span>,</span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a> <span class="st">"-0.07"</span>,</span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a> <span class="st">"+0.25"</span>,</span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a> <span class="st">"-0.84"</span>,</span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a> <span class="st">"+0.32"</span>,</span>
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a> <span class="st">"-0.24"</span>,</span>
<span id="cb3-8"><a href="#cb3-8" aria-hidden="true" tabindex="-1"></a> <span class="st">"-0.97"</span>,</span>
<span id="cb3-9"><a href="#cb3-9" aria-hidden="true" tabindex="-1"></a> <span class="st">"-0.36"</span>,</span>
<span id="cb3-10"><a href="#cb3-10" aria-hidden="true" tabindex="-1"></a> <span class="st">"+1.76"</span>,</span>
<span id="cb3-11"><a href="#cb3-11" aria-hidden="true" tabindex="-1"></a> <span class="st">"-0.36"</span></span>
<span id="cb3-12"><a href="#cb3-12" aria-hidden="true" tabindex="-1"></a>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>For the following questions, provide your answers in a code cell.</p>
<ol type="1">
<li><p>What data type does the vector contain?</p></li>
<li><p>Create two new vectors called <code>my_vec_double</code> and <code>my_vec_int</code> which converts <code>my_vec</code> to Double & Integer types, respectively,</p></li>
<li><p>Create a new vector <code>my_vec_bool</code> which comprises of:</p>
<ul>
<li><code>TRUE</code>if an element in <code>my_vec_double</code> is <span class="math inline">\(\le 0\)</span></li>
<li><code>FALSE</code> if an element in <code>my_vec_double</code> is <span class="math inline">\(\ge 0\)</span></li>
</ul>
<p>How many elements of <code>my_vec_double</code> are greater than zero?</p></li>
<li><p>Sort the values of <code>my_vec_double</code> in ascending order.</p></li>
</ol>
<p><br><br><br><br> <br><br><br><br></p>
<hr>
</section>
<section id="question-3" class="level2">
<h2 class="anchored" data-anchor-id="question-3">Question 3</h2>
<div class="callout-tip callout callout-style-default callout-captioned">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<div class="callout-caption-container flex-fill">
50 points
</div>
</div>
<div class="callout-body-container callout-body">
</div>
</div>
<p>In this question we will get a better understanding of how <code>R</code> handles large data structures in memory.</p>
<ol type="1">
<li>Provide <code>R</code> code to construct the following matrices: <span class="math display">\[
\begin{bmatrix}
1 & 2 & 3\\
4 & 5 & 6\\
7 & 8 & 9\\
\end{bmatrix}
\quad \text{ and } \quad
\begin{bmatrix}
1 & 2 & 3 & 4 & 5 & \dots & 100\\
1 & 4 & 9 & 16 & 25 & \dots & 10000\\
\end{bmatrix}
\]</span></li>
</ol>
<div class="callout-warning callout callout-style-default callout-captioned">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<div class="callout-caption-container flex-fill">
Tip
</div>
</div>
<div class="callout-body-container callout-body">
<p>Recall the discussion in class on how <code>R</code> fills in matrices</p>
</div>
</div>
<p>In the next part, we will discover how knowledge of the way in which a matrix is stored in memory can inform better code choices. To this end, the following function takes an input <span class="math inline">\(n\)</span> and creates an <span class="math inline">\(n \times n\)</span> matrix with random entries.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>generate_matrix <span class="ot"><-</span> <span class="cf">function</span>(n){</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span>(</span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">matrix</span>(</span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">rnorm</span>(n<span class="sc">^</span><span class="dv">2</span>),</span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a> <span class="at">nrow=</span>n</span>
<span id="cb4-6"><a href="#cb4-6" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb4-7"><a href="#cb4-7" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb4-8"><a href="#cb4-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>For example:</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="fu">generate_matrix</span>(<span class="dv">4</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> [,1] [,2] [,3] [,4]
[1,] 0.6398861 0.35207981 -0.6863098 -0.01839937
[2,] 0.7613301 0.72302916 1.5097731 0.48976233
[3,] -0.2608060 0.53979026 -1.0638331 -0.42020192
[4,] 0.6602682 0.08892854 2.3460769 0.03895466</code></pre>
</div>
</div>
<p>Let <code>M</code> be a fixed <span class="math inline">\(50 \times 50\)</span> matrix</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a>M <span class="ot"><-</span> <span class="fu">generate_matrix</span>(<span class="dv">50</span>)</span>
<span id="cb7-2"><a href="#cb7-2" aria-hidden="true" tabindex="-1"></a><span class="fu">mean</span>(M)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.03185503</code></pre>
</div>
</div>
<ol start="2" type="1">
<li>Write a function <code>row_wise_scan</code> which scans the entries of <code>M</code> one row after another and outputs the number of elements whose value is <span class="math inline">\(\ge 0\)</span>. You can use the following <strong>starter code</strong></li>
</ol>
<div class="sourceCode" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>row_wise_scan <span class="ot"><-</span> <span class="cf">function</span>(x){</span>
<span id="cb9-2"><a href="#cb9-2" aria-hidden="true" tabindex="-1"></a> n <span class="ot"><-</span> <span class="fu">nrow</span>(x)</span>
<span id="cb9-3"><a href="#cb9-3" aria-hidden="true" tabindex="-1"></a> m <span class="ot"><-</span> <span class="fu">ncol</span>(x)</span>
<span id="cb9-4"><a href="#cb9-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-5"><a href="#cb9-5" aria-hidden="true" tabindex="-1"></a> <span class="co"># Insert your code here</span></span>
<span id="cb9-6"><a href="#cb9-6" aria-hidden="true" tabindex="-1"></a> count <span class="ot"><-</span> <span class="dv">0</span></span>
<span id="cb9-7"><a href="#cb9-7" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span>(...){</span>
<span id="cb9-8"><a href="#cb9-8" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span>(...){</span>
<span id="cb9-9"><a href="#cb9-9" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(...){</span>
<span id="cb9-10"><a href="#cb9-10" aria-hidden="true" tabindex="-1"></a> count <span class="ot"><-</span> count <span class="sc">+</span> <span class="dv">1</span> </span>
<span id="cb9-11"><a href="#cb9-11" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb9-12"><a href="#cb9-12" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb9-13"><a href="#cb9-13" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb9-14"><a href="#cb9-14" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb9-15"><a href="#cb9-15" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span>(count)</span>
<span id="cb9-16"><a href="#cb9-16" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<ol start="3" type="1">
<li>Similarly, write a function <code>col_wise_scan</code> which does exactly the same thing but scans the entries of <code>M</code> one column after another</li>
</ol>
<div class="sourceCode" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>col_wise_scan <span class="ot"><-</span> <span class="cf">function</span>(x){</span>
<span id="cb10-2"><a href="#cb10-2" aria-hidden="true" tabindex="-1"></a> count <span class="ot"><-</span> <span class="dv">0</span></span>
<span id="cb10-3"><a href="#cb10-3" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb10-4"><a href="#cb10-4" aria-hidden="true" tabindex="-1"></a> ... <span class="co"># Insert your code here</span></span>
<span id="cb10-5"><a href="#cb10-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb10-6"><a href="#cb10-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span>(count)</span>
<span id="cb10-7"><a href="#cb10-7" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<p>You can check if your code is doing what it’s supposed to using the function here<a href="#fn1" class="footnote-ref" id="fnref1" role="doc-noteref"><sup>1</sup></a></p>
<ol start="4" type="1">
<li><p>Between <code>col_wise_scan</code> and <code>row_wise_scan</code>, which function do you expect to take shorter to run? Why?</p></li>
<li><p>Write a function <code>time_scan</code> which takes in a method <code>f</code> and a matrix <code>M</code> and outputs the amount of time taken to run <code>f(M)</code></p></li>
</ol>
<div class="sourceCode" id="cb12"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb12-1"><a href="#cb12-1" aria-hidden="true" tabindex="-1"></a>time_scan <span class="ot"><-</span> <span class="cf">function</span>(f, M){</span>
<span id="cb12-2"><a href="#cb12-2" aria-hidden="true" tabindex="-1"></a> initial_time <span class="ot"><-</span> ... <span class="co"># Write your code here</span></span>
<span id="cb12-3"><a href="#cb12-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">f</span>(M)</span>
<span id="cb12-4"><a href="#cb12-4" aria-hidden="true" tabindex="-1"></a> final_time <span class="ot"><-</span> ... <span class="co"># Write your code here</span></span>
<span id="cb12-5"><a href="#cb12-5" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb12-6"><a href="#cb12-6" aria-hidden="true" tabindex="-1"></a> total_time_taken <span class="ot"><-</span> final_time <span class="sc">-</span> initial_time</span>
<span id="cb12-7"><a href="#cb12-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span>(total_time_taken)</span>
<span id="cb12-8"><a href="#cb12-8" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<p>Provide your output to</p>
<div class="sourceCode" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a><span class="fu">list</span>(</span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a> <span class="at">row_wise_time =</span> <span class="fu">time_scan</span>(row_wise_scan, M),</span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a> <span class="at">col_wise_time =</span> <span class="fu">time_scan</span>(row_wise_scan, M)</span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<p>Which took longer to run?</p>
<ol start="6" type="1">
<li>Repeat this experiment now when:
<ul>
<li><code>M</code> is a <span class="math inline">\(100 \times 100\)</span> matrix</li>
<li><code>M</code> is a <span class="math inline">\(1000 \times 1000\)</span> matrix</li>
<li><code>M</code> is a <span class="math inline">\(5000 \times 5000\)</span> matrix</li>
</ul></li>
</ol>
<p>What can you conclude?</p>
<p><br><br><br><br> <br><br><br><br></p>
<hr>
</section>
<section id="appendix" class="level1">
<h1>Appendix</h1>
<p>Print your <code>R</code> session information using the following command</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb14"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb14-1"><a href="#cb14-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sessionInfo</span>()</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>R version 4.2.2 (2022-10-31)
Platform: x86_64-apple-darwin22.1.0 (64-bit)
Running under: macOS Ventura 13.1
Matrix products: default
BLAS: /usr/local/Cellar/openblas/0.3.21/lib/libopenblasp-r0.3.21.dylib
LAPACK: /usr/local/Cellar/r/4.2.2_1/lib/R/lib/libRlapack.dylib
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
attached base packages:
[1] stats graphics grDevices datasets utils methods base
loaded via a namespace (and not attached):
[1] digest_0.6.31 lifecycle_1.0.3 jsonlite_1.8.4 magrittr_2.0.3
[5] evaluate_0.20 rlang_1.0.6 stringi_1.7.12 cli_3.6.0
[9] renv_0.16.0-53 vctrs_0.5.1 rmarkdown_2.20 tools_4.2.2
[13] stringr_1.5.0 glue_1.6.2 xfun_0.36 yaml_2.3.6
[17] fastmap_1.1.0 compiler_4.2.2 htmltools_0.5.4 knitr_1.41 </code></pre>
</div>
</div>
</section>
<div id="quarto-appendix" class="default"><section id="footnotes" class="footnotes footnotes-end-of-document" role="doc-endnotes"><h2 class="anchored quarto-appendix-heading">Footnotes</h2>
<ol>
<li id="fn1"><p>If your code is right, the following code should evaluate to be <code>TRUE</code></p>
<div class="sourceCode" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a><span class="fu">sapply</span>(<span class="dv">1</span><span class="sc">:</span><span class="dv">100</span>, <span class="cf">function</span>(i) {</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a> x <span class="ot"><-</span> <span class="fu">generate_matrix</span>(<span class="dv">100</span>)</span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a> <span class="fu">row_wise_scan</span>(x) <span class="sc">==</span> <span class="fu">col_wise_scan</span>(x)</span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a>}) <span class="sc">%>%</span> sum <span class="sc">==</span> <span class="dv">100</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<a href="#fnref1" class="footnote-back" role="doc-backlink">↩︎</a></li>
</ol>
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