## Why
Persisting user input received during an active turn currently blocks the next model request. Stores that support background persistence can overlap this checkpoint with inference.
## What changed
- Add `PersistContext::SteeredUserInput` and `allows_background_persistence()` so stores may enqueue these checkpoints, with durability and error reporting enforced by later flush or shutdown operations.
- Use the new context for accepted steered user input and apply the same metadata handling as turn-start persistence.
- Keep tool outputs synchronous, including in mixed input batches, and allow stores to retain synchronous persistence for all contexts.
## Testing
Add gated-store integration tests covering background user-input persistence, synchronous stores, and synchronous tool-output checkpoints. Verify that the next request includes the steered input and waits for persistence when required.
GitOrigin-RevId: c60b7b6c9b483245fd3169306bcf0de248ccdf35
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