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/Runner/InteractiveLaunchSpecService.cs

@@ -4,6 +4,7 @@ using ClaudeDo.Data.Git;
 using ClaudeDo.Data.Models;
 using ClaudeDo.Data.Repositories;
 using ClaudeDo.Worker.Config;
+using ClaudeDo.Worker.External;
 using ClaudeDo.Worker.Planning;
 using ClaudeDo.Worker.Skills;
 using Microsoft.EntityFrameworkCore;
@@ -42,6 +43,7 @@ public sealed class InteractiveLaunchSpecService
     private readonly WorktreeManager _wtManager;
     private readonly GitService _git;
     private readonly string _claudePath;
+    private readonly HandoffRoundTracker _handoffRounds;
 
     public InteractiveLaunchSpecService(
         IDbContextFactory<ClaudeDoDbContext> dbFactory,
@@ -49,7 +51,8 @@ public sealed class InteractiveLaunchSpecService
         ISessionSkillRegistry skillRegistry,
         WorktreeManager wtManager,
         GitService git,
-        WorkerConfig cfg)
+        WorkerConfig cfg,
+        HandoffRoundTracker handoffRounds)
     {
         _dbFactory = dbFactory;
         _skillSeeder = skillSeeder;
@@ -57,6 +60,7 @@ public sealed class InteractiveLaunchSpecService
         _wtManager = wtManager;
         _git = git;
         _claudePath = cfg.ClaudeBin;
+        _handoffRounds = handoffRounds;
     }
 
     /// <summary>Builds a LaunchSpec for opening an interactive session in a task's worktree.
@@ -393,13 +397,19 @@ public sealed class InteractiveLaunchSpecService
         var systemPromptPath = Path.Combine(sessionDir, "system-prompt.md");
         await File.WriteAllTextAsync(systemPromptPath, PromptFiles.ReadOrDefault(phase.PromptKind), ct);
 
-        // A "_final" round differs from its non-final counterpart in exactly one rendered line --
-        // the same system prompt and model drive both -- marking the final round and forbidding a
-        // further rerun, so the chain is bounded structurally rather than by prompt appeal alone.
-        var finalNote = phase.IsFinal
-            ? "\nThis is the FINAL round of this list-handler run -- do not start any further reruns; " +
-              "merge what you can, report the rest, and print the summary."
-            : "";
+        // The round count/cap come from HandoffRoundTracker -- the same instance HandoffMcpTools
+        // registered this handoff against -- never recomputed here, so the brief can't drift from
+        // the server-side cap that actually enforces the chain. A "_final" round adds one more line
+        // forbidding a further rerun, so the chain is bounded structurally rather than by prompt
+        // appeal alone.
+        var round = _handoffRounds.GetState(taskId);
+        var remaining = Math.Max(HandoffRoundTracker.MaxHandoffs - round.Count, 0);
+        var roundInfo = $"Handoff budget: this is round {round.Count} of " +
+            $"{HandoffRoundTracker.MaxHandoffs} possible handoffs for this list-handler run " +
+            $"({remaining} more possible after this one).";
+        if (phase.IsFinal)
+            roundInfo += "\nThis is the FINAL round of this list-handler run -- do not start any further reruns; " +
+                "merge what you can, report the rest, and print the summary.";
 
         var briefPath = Path.Combine(sessionDir, "handoff.md");
         await File.WriteAllTextAsync(briefPath, PromptFiles.Render(PromptKind.MergeHelperHandoff,
@@ -409,7 +419,7 @@ public sealed class InteractiveLaunchSpecService
                 ["repo"] = repoDir,
                 ["handlerTaskId"] = taskId,
                 ["tasks"] = RenderGroupedBrief(briefEntries),
-                ["finalNote"] = finalNote,
+                ["roundInfo"] = roundInfo,
             }), ct);
 
         var resolvedClaude = WindowsTerminalLauncher.Resolve(_claudePath)