Add broadcast input mode for multi-session typing

Broadcast mode sends keystrokes from the active terminal to all other
connected sessions simultaneously. Useful for applying the same
commands across multiple servers.

- Toggle via Ctrl+Shift+B keyboard shortcut or broadcast button in tab bar
- Visual indicator: broadcast button pulses red when active
- Only sends to sessions in 'connected' state
- Notification on toggle
- State stored in app state object

frontend/src/js/state.js

@@ -25,6 +25,9 @@ class State {
         /** @type {boolean} */
         this.secretStoreInitialised = false;
 
+        /** @type {boolean} Whether broadcast mode is active (keystrokes to all sessions). */
+        this.broadcastMode = false;
+
         /** @type {Map<string, Set<Function>>} event name → callbacks */
         this._listeners = new Map();
     }