Customize the theme output for the search box in Drupal

I am trying to change the HTML that is output when I prin开发者_运维百科t $search_box in my page.tpl.php file. I understand the process of overriding something in drupal but I have no idea how to do it for the search box.

This is the only bit of information that I have found that seems related but I don't known how to apply it to solving my problem: http://api.drupalecommerce.org/api/function/search_form/6-4

You should check search-block-form.tpl.php and search-theme-form.tpl.php located in modules/search, as well as their preprocess functions template_preprocess_search_block_form() and template_preprocess_search_theme_form().

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