refactor(background-agent): split manager.ts into focused modules

Extract 30+ single-responsibility modules from manager.ts (1556 LOC):
- task lifecycle: task-starter, task-completer, task-canceller, task-resumer
- task queries: task-queries, task-poller, task-queue-processor
- notifications: notification-builder, notification-tracker, parent-session-notifier
- session handling: session-validator, session-output-validator, session-todo-checker
- spawner: spawner/ directory with focused spawn modules
- utilities: duration-formatter, error-classifier, message-storage-locator
- result handling: result-handler-context, background-task-completer
- shutdown: background-manager-shutdown, process-signal

src/features/background-agent/process-signal.ts

@@ -0,0 +1,19 @@
+export type ProcessCleanupEvent = NodeJS.Signals | "beforeExit" | "exit"
+
+export function registerProcessSignal(
+  signal: ProcessCleanupEvent,
+  handler: () => void,
+  exitAfter: boolean
+): () => void {
+  const listener = () => {
+    handler()
+    if (exitAfter) {
+      // Set exitCode and schedule exit after delay to allow other handlers to complete async cleanup
+      // Use 6s delay to accommodate LSP cleanup (5s timeout + 1s SIGKILL wait)
+      process.exitCode = 0
+      setTimeout(() => process.exit(), 6000)
+    }
+  }
+  process.on(signal, listener)
+  return listener
+}