feat(worker): per-user box status and lifecycle helpers

Centralize per-user box home-path derivation, status inspection, and
lifecycle (start/stop/restart) in a new box.ts so the HTTP routes and the
terminal WS bridge share one source of truth. Add inspectUserBoxStatus to
docker.ts (single `inspect --format` call, all-null/false on non-zero exit)
and resetBox to user-container.ts (clears the idle timer and drops the
registry entry so a stopped box isn't treated as held).

Manual verification (docker required, after Task 4 wires the routes): with
the worker running and a box up,
`curl -H "Authorization: Bearer $TOKEN" localhost:3921/box/status?user=<you>`
shows running:true + image + startedAt; `POST /box/lifecycle {"action":"stop"}`
flips status to running:false; `start` brings it back.

apps/agent-worker/src/runtime/docker.ts

@@ -214,6 +214,42 @@ export const ensureUserContainer = async (args: {
   return name;
 };
 
+// Inspect the per-user box in one shot. Non-zero exit (no such container) yields
+// an all-null/false status. `--memory 0` means unlimited, which the Go template
+// prints as `0`; we normalize that (and any unparseable value) to null.
+export const inspectUserBoxStatus = async (
+  username: string,
+): Promise<{
+  running: boolean;
+  image: string | null;
+  startedAt: string | null;
+  memoryLimitBytes: number | null;
+}> => {
+  const result = await execa(
+    containerRuntime(),
+    [
+      'inspect',
+      '--format',
+      '{{.State.Running}}|{{.Config.Image}}|{{.State.StartedAt}}|{{.HostConfig.Memory}}',
+      userContainerName(username),
+    ],
+    { reject: false, stdin: 'ignore' },
+  );
+  if (result.exitCode !== 0) {
+    return { running: false, image: null, startedAt: null, memoryLimitBytes: null };
+  }
+  const [runningField, imageField, startedAtField, memoryField] = result.stdout
+    .trim()
+    .split('|');
+  const running = runningField === 'true';
+  return {
+    running,
+    image: imageField || null,
+    startedAt: running ? startedAtField || null : null,
+    memoryLimitBytes: Number(memoryField) || null,
+  };
+};
+
 const shellQuote = (value: string) => `'${value.replaceAll("'", "'\\''")}'`;
 
 // The marker is embedded literally into a `# comment` line and into a `pgrep`