php - Passing user entered variable among multiple pages

one text

Basically, what I want to do is to have the rep/user enter their name on the first page they see, and it would insert their entered name into the script(s) depending on which PHP page they have selected from the side-menu. It was mentioned to use a $_SESSION variable...but I'm a bit confused on this as would it then require passing the variable from a php to a php first so that it can be called? Currently my pages are as such:

Initial name entry page for script:

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
<head>
    <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
    <title> Simple HTML Form</title>
    <style type="text/css" title="text/css" media="all">
    label {
        font-weight: bold;
        color: #300ACC;
        }
    </style>
</head>
<body>
<!-- Script 2.1 form.html -->

<form action="handle_CHIS.php" method="post">

    <fieldset><legend>Enter your Name/Phone Alias in the form below: </legend>
    
    <p><label>Name:  <input type="text" name="name" size="20" maxlength="40" /></label></p>
    
    </fieldset>
    
    <p align="center"><input type="submit" name="submit" value="Submit My Information" /></p>
    
    
</form>

</body>
</html>

Then the script (a php page) would then use the entered Name/Phone Alias

<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en" lang="en">
    <head>
        <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
        <title>California Health Survey</title>
        <link rel="stylesheet" href="style.css">
        <link rel="stylesheet" href="https://use.fontawesome.com/releases/v5.3.1/css/all.css">
    </head>
    <body>
    <?php

    if ( !empty($_POST['name']) ) {
        echo "<p>Hello, my name is <font color=#FF0000><B>{$_POST['name']}</B></font>. I am calling for <font color=#000080><B>UCLA</B></font>.  We are doing a scientific study about health in California. <br />
Have I reached you at _____?</p>\n";
    }  
    ?>
    </body>

    <div class="middle">
      <div class="menu">
        <li class="item" id='profile'>
          <a href="#profile" class="btn">If Necessary</a>
          <div class="smenu">
            <a href="#">Our questions are for research purposes only and your answers are strictly confidential.)</a>
          </div>
        </li>

        <li class="item" id="reassurance">
          <a href="#reassurance" class="btn"></i>Reassurance</a>
          <div class="smenu">
            <a href="#">I want to assure you we are not selling anything.</a>
          </div>
        </li>
    
    </div>

</html>

How would I do this so that EVERY study would not require an individual HTML and PHP page, but rather just ONE html for Name/Alias entry which would then populate into each script page? Would it require 2 PHP pages so that the first's "form action" would point to the second? I've only seen it done from one page to another page. Even the information I've come across has only been to pass it from one page to one other by referencing that specific page. I want to have the user be able to enter their name and have it populate in ALL scripts individually when clicked on my scripts.

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