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Maintain the current hash associated each Page's state and use that rather than
querying location.hash directly. This fixes several long-standing issues with
the settings search page and the back/forward list:

- Opening an overlay from a search page no longer keeps the fragment in the
  URL.
- Closing an overlay no longer loses the search in the URL.
- Navigating back from one search page to another actually works.
- Navigating back from the search page doesn't clear the search box.

Add tests for this behavior.

BUG=410204

Review URL: https://codereview.chromium.org/555163007

Cr-Commit-Position: refs/heads/master@{#294614}