AI optimized ‘Search’ function within website

Topic: Using AI for enhanced website Search performance: From the standpoint of enhancing a website’s ‘Search’ function, what would be an optimal way of establishing logic (or code) to first review the site for content applicable to the Search entry, but if not present within the site, would then go out to a chatbot such as Bard, ChatGPT, or Claude to find and present an applicable response, and finally retain the Search history for further site development or enhancement?