feat(eve): add Exa web search support

[kesku]

Description

Closes #688.

Adds Exa as a web_search backend for agents configured with an AI Gateway string model ID.

Exa is available as a provider-managed AI Gateway tool so Eve can inject gateway.tools.exaSearch() without requiring developers to define a custom tool, supply a separate Exa API key, or add new configuration.

Model behavior is unchanged: supported providers continue to use their native web search tools, and user-defined web_search overrides continue to take precedence.

Also adds gateway output schema, unit coverage for backend resolution and tool injection, and a E2E eval.

How did you test your changes?

pnpm lint
pnpm typecheck
pnpm --filter eve exec vitest run \
  --config vitest.unit.config.ts \
  src/harness/provider-tools.test.ts \
  src/harness/tools.test.ts \
  src/harness/tool-loop.test.ts

Also tested with a production AI_GATEWAY_API_KEY end to end.

PR Checklist

• I linked an issue with prior discussion confirming this change is wanted
• I ran the relevant checks from CONTRIBUTING.md
• I added tests and documentation where relevant
• I added a changeset if this touches the published eve package
• DCO sign-off passes for every commit (git commit --signoff)

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