feat(web): use a global WebSocket for Pico chat sessions (#1507)

- centralize Pico chat connection and session state in a shared store
- move chat lifecycle control out of usePicoChat
- hydrate and restore the active session across the app

web/frontend/src/routes/__root.tsx

@@ -1,9 +1,15 @@
 import { Outlet, createRootRoute } from "@tanstack/react-router"
 import { TanStackRouterDevtools } from "@tanstack/react-router-devtools"
+import { useEffect } from "react"
 
 import { AppLayout } from "@/components/app-layout"
+import { initializeChatStore } from "@/lib/pico-chat-controller"
 
 const RootLayout = () => {
+  useEffect(() => {
+    initializeChatStore()
+  }, [])
+
   return (
     <AppLayout>
       <Outlet />