Add Parallel research tool to Pi

[georgeatparallel]

This adds web_research to the existing Parallel Pi extension, so Pi can hand off a complete research question and get an answer with sources. It uses the existing login and does not need another package or a model switch. Search and Fetch keep their existing jobs: finding sources and reading URLs.

For a follow-up, Pi can pass the latest returned response ID as previous_response_id to reuse that investigation's context. Unrelated questions start fresh. IDs appear in the tool's visible result, including long-answer previews, so Pi can use them without storing research sessions locally. Missing context and retention restrictions return errors without silently retrying as new research.

Research defaults to medium effort and preserves the answer's formatting and cited passages. Large results keep the complete report in a private temporary file. Calls have a 120-second client deadline, cancellation, and no automatic retries. The extension does not automatically send Pi's conversation history or local files.

Validation: 237 deterministic tests passed, including 136 Pi tests, plus lint, formatting, type checks, builds, and packaging. The continuation contract was checked against the Responses API and Pi source. Forty-six packed-extension cases passed through the actual Pi runtime, including concurrent investigations and session restore. A live two-turn check confirmed that Pi passed the returned ID and the API recalled an earlier constraint that was absent from the follow-up query. Fourteen unrelated credentialed integration tests were excluded from the deterministic suite.

The package version stays at 1.2.0, with no added dependency or release.