Append Message To Task

Append the next user message to an existing task and kick off its next turn.

Phase 1 multi-turn model is task-centric: subsequent user inputs extend the same task_id rather than creating a new task or a new conversation_id. This endpoint:

1. Validates the path task_id exists and belongs to the key-bound agent (404 task_not_found otherwise).
2. Validates body.agent_id matches the key-bound agent (404 agent_not_found otherwise).
3. Rejects the call with 409 task_busy if the task is currently RUNNING -- the SDK client should poll GET /v1/chat/tasks/{id} until status leaves RUNNING and retry.
4. Otherwise persists the new user message to task_chat_messages, updates task.input to record this turn's input, and kicks off the next background turn via the same helper POST uses.

Args: task_id: Path parameter; the target task's primary key. request: Validated :class:AppendMessageRequest. message.content is guaranteed non-empty by Pydantic. authed: (Agent, AgentApiKey) from the auth dependency. db: SQLAlchemy session.

Returns: :class:AppendMessageResponse with the task identity and an accepted_at timestamp.

Raises: V1ApiError 401: missing / invalid / revoked key. V1ApiError 404: task not found OR not owned by the agent OR body.agent_id doesn't match the bound agent. V1ApiError 409: task_busy -- task currently RUNNING. 500: any other unexpected error (V1 envelope via global handler).