fix: prevent agent participants getting stuck in private chat

[m9h]

Summary

Fixes #1011 and #938 — two related bugs that cause private chat stages to hang indefinitely.

Bug 1: Spinner hangs when mediator returns shouldRespond: false (#938)

When a mediator's structured output includes shouldRespond: false, no message is written to Firestore. The frontend has no signal, so the "Waiting for a response..." spinner hangs for 120 seconds until the timeout fires.

Bug 2: Agent participants get stuck in private chat (#1011)

When a mediator reaches maxResponses or returns readyToEndChat: true with an empty message, no new chat document is created. Agent participants only respond to UserType.MEDIATOR messages, so they never get triggered and the experiment is stuck forever.

Root cause

Both bugs share the same root cause: when a mediator stops responding (for any reason), no Firestore document is written, so neither the frontend nor agent participants receive a signal.

Fix

When a mediator can no longer respond, send a system message ("<Name> has left the chat.") to the private chat collection. This:

• Stops the frontend spinner — the system message's senderId differs from the participant's publicId, breaking the isWaitingForResponse condition
• Unblocks agent participants — the trigger now allows agents to respond to UserType.SYSTEM messages (in addition to UserType.MEDIATOR), so they can set readyToEndChat and advance
• Uses trigger log deduplication (left-chat-{publicId}) to prevent infinite loops (system message → trigger → mediator can't respond → system message → ...)
• Skips mediator responses to system messages — only agent participants should react to "left the chat" signals, not mediators

Also moves the empty-text failure check (!response.text) to after structured output extraction, so shouldRespond: false and readyToEndChat: true with empty text aren't incorrectly treated as API failures.

Changes

File	Change
functions/src/chat/chat.agent.ts	Send "left the chat" system message when mediator hits maxResponses or returns shouldRespond: false; reorder empty-text check
functions/src/chat/chat.utils.ts	Add sendSystemPrivateChatMessage() utility
functions/src/triggers/chat.triggers.ts	Skip mediator responses to system messages; allow agent participants to respond to system messages

Test plan

• Test with mediator configured with maxResponses: 3 — verify system message appears after 3rd response and agent advances
• Test with mediator returning shouldRespond: false via structured output — verify spinner stops immediately and system message appears
• Test normal private chat flow (no shouldRespond/maxResponses) — verify no regression
• Test with human participant (not agent) — verify "left the chat" message appears and spinner stops
• Verify no infinite loop: system message should only be sent once per mediator per stage (check trigger logs)

🤖 Generated with Claude Code

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[rasmi]

Hi @m9h -- thanks for the PR! It seems to have a lot of unrelated commits?

[m9h]

Sorry about that @rasmi. Those weren't supposed to be included. I will clean this up as soon as I can get back to it.

[cjqian]

@rasmi is this related to #1011 ?