[KirschBluteX]

• I have read and reviewed the documentation changes to the best of my ability.
• If the change is significant, I have run the documentation site locally and confirmed it renders as expected.

Summary of changes

• Document the TaskToolSet background lifecycle added by OpenHands/software-agent-sdk#4503.
• Show the blocking default and explicit run_in_background=True opt-in.
• Document task_output, task_stop, task states, cooperative cancellation, bounded close behavior, and exactly-once metrics settlement.
• Clarify that task IDs are process-local and are not restored after a process restart; active resume is rejected.

Validation

• git diff --check passed.
• The documented imports and lifecycle match the SDK implementation and its 165 focused/related tests.
• Full-checkout Mint and desktop/mobile render checks were run previously. The final sparse checkout reports baseline missing-file/navigation warnings for files outside this PR; those files are unchanged here.

The API details were checked against the SDK TaskToolSet definitions and manager: output reads are bounded by a finite timeout, stop is cooperative through the child LocalConversation, and task IDs remain process-local control handles.

[enyst]

I'm an AI agent based on Opus 5.5, helping Engel Nyst (@enyst) with project work.

I checked this against SDK #4503 at its current head (97a18e4). run_in_background, task_output (block, finite timeout 0–3600), task_stop, the five statuses, and the rejection of a resume while a task is still active all match (definition.py#L65, definition.py#L128-L144, manager.py#L398).

• #4503 is still open, so this should wait until it merges. That PR has changed quite a bit since Aug 18, so a quick re-check after it lands would help.
• The rewrite removes the ready-to-run example code. The old block pointed at 40_task_tool_set.py, which no longer exists, so it was out of date. Please point the block at the current file instead of deleting it (see the inline comment).
• The background snippet runs a separately created executor. In normal use, the parent LLM calls task(run_in_background=true), task_output, and task_stop itself. I'd say that first and label the snippet as direct/programmatic use.

[enyst]

Please keep the example code here and point it at the current file (it's 41_, not 40_). Put this above the RunExampleCode line:

```python icon="python" expandable examples/01_standalone_sdk/41_task_tool_set.py
<current contents of examples/01_standalone_sdk/41_task_tool_set.py>
```

Blocks that carry an example path are synced from the SDK (see AGENTS.md), so the correct path is what keeps the code current.