chore: migrate contexts to stores and refresh barrel exports

- Move state management from `src/contexts` to `src/stores` (zustand-based)
- Update barrelsby config to include new stores directory
- Add bottom-sheet and auth-background components
- Auto-open browser on dev server ready in VS Code launch config
- Refresh generated barrel index files

src/app/chat/components/chat-screen.tsx

@@ -22,7 +22,7 @@ import { useEffect, useRef, useState } from "react";
 import Image from "next/image";
 
 import { useChatDispatch, useChatState } from "@/stores/chat/chat-context";
-import { GuestChatQuota } from "@/stores/chat/chat-types";
+import { GuestChatQuota } from "@/stores/chat";
 
 import { MobileShell } from "@/app/_components/core/mobile-shell";