Add web search and fetch backend selection

[antonvishal]

Why

Let users choose how fx searches the web and retrieves pages without editing environment variables, while keeping domain permission rules enforceable.

What

• Add a two-stage /web picker, independent Settings controls, and fx web --json.
• Support Exa, Browserbase, Parallel, Perplexity, and Tako search, plus local HTTP and Browserbase fetch.
• Save selections in profile settings with environment overrides and inherit them in subagents across CLI, TUI, and ACP.
• Hide Browserbase fetch when domain permission rules require local admission, except in full-access mode.
• Configure Tako result counts and domain filters through its source-specific options.

How

Share typed backend selections across entry points, filter provider execution through the permission policy, and serialize each backend's supported options.

Build, formatting, the full Zig unit suite, CLI/search tests, gateway regressions, and real-terminal picker/settings tests pass locally.

[vercel-security-reviewer]

Security review details

[antonvishal]

@fazxes AI Gateway now natively supports multiple web search tools.

Would love your review here.