Wanted: Search Results on 404 page

I’d like to add search results to a 404 page on an EE site.

My thought is to extract the URL segments, and then send them through the search process as if they were keywords entered into a search form, then display the search results.

However, all I can find in the EE user guide is instructions on how to set up a form, and how to display search results, but not a way to send keywords to the search form from the backend without a form submission.

I’d like this to happen automatically, so that the 404 page can display the search results, instead of requiring the user to click somewhere, or hack it with an AJAX form submission.

Anyone have any ideas on how this could be accomplished?

Thank you!