Enqueue the next input after the current turn

Queues message as the session's next turn in the same engine conversation, after every turn already queued. Only an active session takes input; the other admission states answer the 409 codes below. The 202 means the input is stored under a new sequential turn_id, not that it ran: follow the turn with GET /v1/sessions/{id}/turns/{turn_id} or on the event stream. A session over its budget still queues input but runs no new turn (attention.code: BUDGET_EXCEEDED). The receipt settles with the turn: succeeded when it completes, failed when it fails or is interrupted or cancelled, unknown when its outcome is unknown. A caller who sees the session but may not use its private agent (GET /v1/agents/{id}/usage) is refused with 403 FORBIDDEN, reason_code agent_private, and no one to ask; a retry of an input taken before they lost the agent still gets its stored answer. A sender whose caller limits (GET /v1/agents/{id}/caller-limits) leave out something the session loads, after the limit the session was created under took its part away, is refused with 403 FORBIDDEN, reason_code caller_limited and details.dimension (tool, mcp_server, repository or credential); a rule change counts from the next input.

Scope: sessions:write (API key or cookie session). Idempotency-Key: required.