feat(app): glassmorphism chat UI with widget embed system

- Glassmorphism chat window matching Proux design rules (glass layers,
  gradient backgrounds, glow effects, blur intensities, inner glows)
- Conversation ID displayed in header, side-switching (left/right layout)
- Chat store with Pinia: conversations, messages, streaming state
- OpenRouter AI integration with SSE streaming (Llama 4 Maverick free model)
- Chat components: ChatWindow, ChatHeader, ChatInput, ChatMessage,
  StreamingDots (typing indicator)
- Embeddable widget system: floating action button (FAB) + modal popup
  with scale-in animation, side-aware positioning
- Widget demo page (/widget-demo) showing AIUI embedded in a mock "Acme App"
  with documentation of 4 integration methods: script tag, web component,
  npm package, and iframe
- Theme composable with dark/light mode, system preference detection
- Custom CSS: glass variants (subtle/medium/strong/light/dark), glow effects,
  gradient text, animation keyframes, thin scrollbar
- Responsive: mobile-first, 44px touch targets, dvh viewport

Made-with: Cursor

packages/app/src/main.ts

@@ -12,6 +12,11 @@ const router = createRouter({
       name: 'chat',
       component: () => import('./pages/ChatPage.vue'),
     },
+    {
+      path: '/widget-demo',
+      name: 'widget-demo',
+      component: () => import('./pages/WidgetDemoPage.vue'),
+    },
   ],
 })