### Summary
- Parse all `web_search` tool actions (`search`, `find_in_page`,
`open_page`).
- Previously we only parsed + displayed `search`, which made the TUI
appear to pause when the other actions were being used.
- Show in progress `web_search` calls as `Searching the web`
- Previously we only showed completed tool calls
<img width="308" height="149" alt="image"
src="https://github.com/user-attachments/assets/90a4e8ff-b06a-48ff-a282-b57b31121845"
/>
### Tests
Added + updated tests, tested locally
### Follow ups
Update VSCode extension to display these as well
**Goal**: Prevent response.failed events with `invalid_prompt` from
being treated as retryable errors so the UI shows the actual error
message instead of continually retrying.
**Before**: Codex would continue to retry despite the prompt being
marked as disallowed
**After**: Codex will stop retrying once prompt is marked disallowed
- capture the header from SSE/WS handshakes, store it per
ModelClientSession using `Oncelock`, echo it on turn-scoped requests,
and add SSE+WS integration tests for within-turn persistence +
cross-turn reset.
- keep `x-codex-turn-state` sticky within a user turn to maintain
routing continuity for retries/tool follow-ups.
