feat: live tab awareness for the Terminal pane

claude in the PTY now has continuous tab-aware context. Three pieces:

1. Live state files. background.js listens to chrome.tabs.onActivated /
   onCreated / onRemoved / onUpdated (throttled to URL/title/status==
   complete so loading spinners don't spam) and pushes a snapshot. The
   sidepanel relays it as a custom event; sidepanel-terminal.js sends
   {type:"tabState"} text frames over the live PTY WebSocket.
   terminal-agent.ts writes:
     <stateDir>/tabs.json          all open tabs (id, url, title, active,
                                   pinned, audible, windowId)
     <stateDir>/active-tab.json    current active tab (skips chrome:// and
                                   chrome-extension:// internal pages)
   Atomic write via tmp + rename so claude never reads a half-written
   document. A fresh snapshot is pushed on WS open so the files exist by
   the time claude finishes booting.

2. New $B tab-each <command> [args...] meta-command. Fans out a single
   command across every open tab, returns
   {command, args, total, results: [{tabId, url, title, status, output}]}.
   Skips chrome:// pages; restores the originally active tab in a finally
   block (so a mid-batch error doesn't leave the user looking at a
   different tab); uses bringToFront: false so the OS window doesn't
   jump on every fanout. Scope-checks the inner command BEFORE the loop.

3. --append-system-prompt hint at spawn time. Claude is told about both
   the state files and the $B tab-each command up front, so it doesn't
   have to discover the surface by trial. Passed via the --append-system-
   prompt CLI flag, NOT as a leading PTY write — the hint stays out of
   the visible transcript.

Tests:
- browse/test/tab-each.test.ts (new) — registration + source-level
  invariants (scope check before loop, finally-restore, bringToFront:false,
  chrome:// skip) + behavior tests with a mock BrowserManager that verify
  iteration order, JSON shape, error handling, and active-tab restore.
- browse/test/terminal-agent.test.ts — three new assertions for
  tabState handler shape, atomic-write pattern, and the
  --append-system-prompt wiring at spawn.

Verified live: opened 5 tabs, ran $B tab-each url against the live
server, got per-tab JSON results back, original active tab restored
without OS focus stealing.

extension/sidepanel.js

@@ -1039,4 +1039,13 @@ chrome.runtime.onMessage.addListener((msg) => {
     inspectorPickerActive = false;
     inspectorPickBtn.classList.remove('active');
   }
+  // browserTabState: full snapshot of all open tabs + the active one,
+  // pushed by background.js on chrome.tabs events. We forward it as a
+  // custom event so sidepanel-terminal.js can relay to terminal-agent.ts.
+  // Result: claude's <stateDir>/tabs.json + active-tab.json stay live.
+  if (msg.type === 'browserTabState') {
+    document.dispatchEvent(new CustomEvent('gstack:tab-state', {
+      detail: { active: msg.active, tabs: msg.tabs, reason: msg.reason },
+    }));
+  }
 });