[GitHub bot integration] Add an ability to sequentially update a single branch

First requested in anthropics/claude-code#4790

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Right now, the Claude bot user creates a new `claude/*` branch every time a user requests it some changes, even when we do it in the same issue/PR or if we ask it to update a specific branch (even one that Claude itself created).

However, I believe it is contrary how Claude Code (CLI) works: using CC we can work on the same (currently checked out) branch until we consider the work done, then commit and push the changes.

More over, current approach with isolated branches creates zillion unnecessary branches, when Claude bot could just create separate commits for subsequent change requests.