## Summary
- Adds chat mode persistence to chats so Ask, Build, and Plan are
remembered per conversation.
- Resolves effective chat mode for chat creation and streaming paths,
including fallback metadata for the UI.
- Updates chat mode hooks, selectors, tab flows, and coverage for
persistence behavior.

## Test plan
- npm run fmt && npm run lint:fix && npm run ts
- npm test
- npm run build
- PLAYWRIGHT_HTML_OPEN=never npm run e2e -- e2e-tests/chat_mode.spec.ts

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Co-authored-by: Will Chen <7344640+wwwillchen@users.noreply.github.com>