Add File Browser to Rapid Diffs

We should add a file browser to Rapid Diffs. In order to do that we should:

1. Change existing file browser width storage from LocalStorage to Cookies. This is needed so we render file browser at a proper width on the server and won't cause layout shift on the client if they resized the file browser.
2. Refactor file browser Vue components to work with state that currently exists in Rapid Diffs. Ideally, we should extract file browser state into a separate Pinia store and use only that.
3. Actually render file browser on the page with the correct data.
4. Handle events from the file browser: react to clicks on files and scroll to appropriate files.
5. Handle loading state for streamed files. We should indicate that a file has not loaded yet in the file browser.