feat(worker): surface the handoff round budget in the merge-helper handoff brief

A handler session only learned the chain was capped when handoff_list_handler
threw at the end of its work, stranding open tasks. The round count already
lived in HandoffRoundTracker the whole time -- it just never reached the
brief. InteractiveLaunchSpecService now reads HandoffRoundTracker.GetState
(read-only, doesn't advance the round) and renders round N of MaxHandoffs
plus the remaining count into the handoff brief, replacing the old one-line
finalNote. The rejection error also names the next action (start a new
handler run) instead of just failing.

src/ClaudeDo.Worker/External/HandoffRoundTracker.cs

@@ -5,8 +5,8 @@ namespace ClaudeDo.Worker.External;
 
 // Bounds the list-handler wait<->merge handoff chain server-side, so a model that ignores the
 // cooperative "this is the FINAL round" line in handoff.md (InteractiveLaunchSpecService's
-// finalNote) cannot loop wait->merge->wait->merge forever, each round spawning a fresh ConPTY
-// session. Keyed by handler task id -- a "Let Claude handle it" run always creates a fresh
+// roundInfo, read back via GetState) cannot loop wait->merge->wait->merge forever, each round
+// spawning a fresh ConPTY session. Keyed by handler task id -- a "Let Claude handle it" run always creates a fresh
 // handler task (InteractiveLaunchSpecService.CreateMergeHelperTaskAsync), so the counter is
 // per-run by construction: no reset logic needed, and a later run on the same list starts at 0
 // on its own. In-memory only, same tradeoff as TaskRunTokenRegistry -- a worker restart mid-run
@@ -26,6 +26,17 @@ public sealed class HandoffRoundTracker
 
     public sealed record Result(bool Rejected, bool Coerced, string EffectivePhase, string? Error);
 
+    /// <summary>Read-only snapshot for a handler task -- does NOT register a round. Used to report
+    /// the current round/remaining budget (e.g. in the handoff brief) without mutating state that
+    /// only Register should advance. Count is 0 for a task that hasn't handed off yet.</summary>
+    public (int Count, bool Sealed) GetState(string taskId)
+    {
+        if (!_states.TryGetValue(taskId, out var state))
+            return (0, false);
+        lock (state)
+            return (state.Count, state.Sealed);
+    }
+
     public Result Register(string taskId, string nextPhase)
     {
         var state = _states.GetOrAdd(taskId, _ => new TaskState());
@@ -33,7 +44,8 @@ public sealed class HandoffRoundTracker
         {
             if (state.Sealed)
                 return new Result(true, false, nextPhase,
-                    $"Task {taskId} already completed its merge_final round -- no further handoff is allowed.");
+                    $"Task {taskId} already completed its merge_final round -- no further handoff is allowed. " +
+                    "Start a new handler run (\"Let Claude handle it\") on this list instead.");
 
             state.Count++;
             var isFinal = nextPhase is MergeHelperPhase.WaitFinal or MergeHelperPhase.MergeFinal;