fix(llm): ensure Google chats end with a user turn

[swayamg20]

Summary

• decide Google dummy-user injection from the final serialized turn instead of the loop's last observed role
• preserve a trailing user turn when an empty interrupted message is omitted from the serialized payload
• add a regression test for the barge-in sequence reported in google: 400 'Requests ending with a model turn are not supported' after barge-in on gemini-3.6-flash — retried in a loop, agent goes silent #2108

This PR only fixes trailing-turn serialization. It does not change Google API error retry classification.

Fixes #2108

Testing

• pnpm exec vitest run agents/src/llm/provider_format/google.test.ts (27 passed)
• pnpm exec vitest run agents/src --reporter=dot (1,586 passed, 5 skipped)
• pnpm --filter @livekit/agents typecheck
• pnpm build (40 packages)
• ESLint and Prettier on changed files
• pnpm exec changeset status

[changeset-bot]

🦋 Changeset detected

Latest commit: b40af79

The changes in this PR will be included in the next version bump.

This PR includes changesets to release 39 packages

Name	Type
@livekit/agents	Patch
@livekit/agents-plugin-anam	Patch
@livekit/agents-plugin-anthropic	Patch
@livekit/agents-plugin-assemblyai	Patch
@livekit/agents-plugin-azure	Patch
@livekit/agents-plugin-baseten	Patch
@livekit/agents-plugin-bey	Patch
@livekit/agents-plugin-cartesia	Patch
@livekit/agents-plugin-cerebras	Patch
@livekit/agents-plugin-deepgram	Patch
@livekit/agents-plugin-did	Patch
@livekit/agents-plugin-elevenlabs	Patch
@livekit/agents-plugin-fishaudio	Patch
@livekit/agents-plugin-google	Patch
@livekit/agents-plugin-hedra	Patch
@livekit/agents-plugin-hume	Patch
@livekit/agents-plugin-inworld	Patch
@livekit/agents-plugin-krisp	Patch
@livekit/agents-plugin-lemonslice	Patch
@livekit/agents-plugin-liveavatar	Patch
@livekit/agents-plugin-livekit	Patch
@livekit/agents-plugin-minimax	Patch
@livekit/agents-plugin-mistral	Patch
@livekit/agents-plugin-mistralai	Patch
@livekit/agents-plugin-neuphonic	Patch
@livekit/agents-plugin-openai	Patch
@livekit/agents-plugin-perplexity	Patch
@livekit/agents-plugin-phonic	Patch
@livekit/agents-plugin-protoface	Patch
@livekit/agents-plugin-resemble	Patch
@livekit/agents-plugin-rime	Patch
@livekit/agents-plugin-runway	Patch
@livekit/agents-plugin-sarvam	Patch
@livekit/agents-plugin-silero	Patch
@livekit/agents-plugin-soniox	Patch
@livekit/agents-plugin-tavus	Patch
@livekit/agents-plugins-test	Patch
@livekit/agents-plugin-trugen	Patch
@livekit/agents-plugin-xai	Patch

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[devin-ai-integration]

✅ Devin Review: No Issues Found

Devin Review analyzed this PR and found no bugs or issues to report.