Google's John Mueller said it would be "extremely rare that Googlebot would submit a form" on your web site. He said Google did this more often in the old days, espesially on governmental web sites where there was no site navigation and the only way to get to public content was to use a search box on the site. But nowadays most sites have good navigation through links and this is rare....
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Google: GoogleBot Would Rarely Ever Submit A Form On Your Website
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