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/spawner/parent-directory-resolver.ts

@@ -0,0 +1,21 @@
+import type { OpencodeClient } from "../constants"
+import { log } from "../../../shared"
+
+export async function resolveParentDirectory(options: {
+  client: OpencodeClient
+  parentSessionID: string
+  defaultDirectory: string
+}): Promise<string> {
+  const { client, parentSessionID, defaultDirectory } = options
+
+  const parentSession = await client.session
+    .get({ path: { id: parentSessionID } })
+    .catch((error: unknown) => {
+      log(`[background-agent] Failed to get parent session: ${error}`)
+      return null
+    })
+
+  const parentDirectory = parentSession?.data?.directory ?? defaultDirectory
+  log(`[background-agent] Parent dir: ${parentSession?.data?.directory}, using: ${parentDirectory}`)
+  return parentDirectory
+}