Google is what we use to search our sites because it the best tool out there.
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The search form on your website is an application powered by Google. It displays results for your website only once Google crawls and indexes the pages.

You will see pages from your old site just until Google drops the old site pages and re-index's your new one (we have no control over the that and how long it will take). The great thing about Google is that is can actually search inside documents for key words.

You site search will start working once your site is live.