Tune documentation search capabilities

## Summary

This is a meta issue for adding and improving the search capabilities of V3.

The general idea at this point is to use the same search that the web team uses. The web team will give us access to their instance and allow us to co-own it with them. This allows us to improve the global search experience while still _owning_ docs search. We would use our own custom header, and not the website header.

## Action items

Not finalized yet. Thanks @reakaleek for these:

- [ ] Figure out what can be enhanced in current search
- [ ] Keep web team with html structure, so global search can still search for it
- [ ] Figure out build time indexing for search
- [x] Action setup staging /docs
- [ ] Test what performs better, fastly cache or cloudfront cache

Notes from @Mpdreamz

> * Cache with fastly makes the most sense for performance. I had hoped it was served from a forward proxy within aws infra so latency is minimal (since request to s3 stay internal).
> * We should avoid injecting html at pageload e.g the [elastic.co](http://elastic.co/) header..., I am sure we can mimic the UI and do requests to https://www.elastic.co/api/autosuggest?q=hello&locale=en
> * I think it makes sense for us to pre-index all the docs content the way we want to in their Elastic Cloud instance ahead of time. We can work together to make their API and search results expose/behave we'd (all hopefully) want it to behave.

### Post 9.0:
- [ ] Accept natural language input https://github.com/elastic/docs-builder/issues/44