Show live background task counts

SUBAGENTS.md

@@ -89,9 +89,11 @@ transcript is still being written to and its process is the session's to stop.
    Since Claude Code 2.1.261, `background_tasks_changed { tasks: [...] }` is
    the authoritative level beside those edges: its set replaces the previous
    set, so a missed terminal edge cannot leave a subagent running forever. Its
-   ids are deliberately not correlated with the edge stream; the useful claim
-   here is whether the set is empty. The edges still carry mapping, outcome
-   and closing summary. On adoption the driver sends a repeated `initialize`,
+   ids are deliberately not correlated with the edge stream; the useful claims
+   here are whether the set is empty and its measured size. The session API and
+   stream expose that size as `backgroundTasks`, which the phone draws beside
+   the status without pretending those tasks are subagents. The edges still
+   carry mapping, outcome and closing summary. On adoption the driver sends a repeated `initialize`,
    which makes a current CLI send the full set; an older CLI accepts it and sends no level,
    leaving the edge-based path unchanged. A snapshot is reconciled immediately
    when the persisted parent status proves it is between turns, and otherwise