feat: setup full Next.js toolchain and core dependencies

- Add runtime deps: react-query, react-hook-form+zod, zustand, ky, openapi-fetch,
  better-auth, react-markdown, shiki, recharts, next-themes, nuqs, sonner, cmdk,
  vaul, embla-carousel-react, motion, date-fns, tailwind-merge, clsx, cva
- Add dev tooling: vitest, testing-library, playwright, prettier(+tailwind plugin),
  husky, lint-staged, jsdom
- Wire providers (QueryClient, Theme, NuqsAdapter, Tooltip, Sonner) into layout
- Add vitest/playwright/prettier configs + lint-staged pre-commit hook
- Add lib/query-client + lib/http (ky) helpers and sample unit test
- Fix react-hooks/set-state-in-effect lint errors (use-mobile useSyncExternalStore,
  carousel async init)
- Remove committed Next-generated types/ duplicates (regenerated under .next/types)

.env.example

@@ -0,0 +1,11 @@
+# Better Auth
+# Generate with: bunx @better-auth/cli secret
+BETTER_AUTH_SECRET=
+# Full absolute URL of the deployed app, e.g. https://example.com
+BETTER_AUTH_URL=
+
+# Optional: database URL for better-auth (Kysely/Drizzle adapter)
+# DATABASE_URL=postgres://user:pass@localhost:5432/db
+
+# Optional: OpenAPI document URL used by openapi-fetch (generate client from it)
+# OPENAPI_URL=/api/openapi.json