Add Vercel Web Analytics integration (#2)

# Vercel Web Analytics Integration

Successfully installed and configured Vercel Web Analytics for the render-modes Next.js project.

## Implementation Summary

### 1. Documentation
- Fetched the latest installation instructions from the official Vercel documentation (https://vercel.com/docs/analytics/quickstart)
- Confirmed this is a Next.js App Router project, which requires the `@vercel/analytics/next` import

### 2. Package Installation
- Installed `@vercel/analytics` package version ^2.0.1 using npm
- Updated package.json to include the new dependency
- Updated package-lock.json with the new dependency tree

### 3. Code Changes
**Modified: app/layout.tsx**
- Added import: `import { Analytics } from "@vercel/analytics/next";`
- Added `<Analytics />` component to the root layout before the closing `</body>` tag
- Preserved all existing code structure, fonts, metadata, and components

### 4. Verification
All quality checks passed successfully:
- ✅ TypeScript type checking (`npm run typecheck`)
- ✅ ESLint linting (`npm run lint`)
- ✅ Production build (`npm run build`)
- ✅ Prettier formatting (`npm run format:check`)

## Files Changed
- **app/layout.tsx** - Added Analytics import and component
- **package.json** - Added @vercel/analytics dependency
- **package-lock.json** - Updated with new dependency tree

## Next Steps
Once deployed to Vercel with Analytics enabled in the dashboard:
1. The Analytics component will automatically track page views
2. Verify by checking browser Network tab for requests to `/<unique-path>/view`
3. View analytics data in the Vercel dashboard under the Analytics section

## Implementation Notes
- The Analytics component was placed at the end of the body tag as recommended by Vercel documentation
- The implementation follows Next.js App Router best practices
- No breaking changes were introduced - all existing functionality remains intact
- The component is lightweight and won't impact page load performance

Co-authored-by: Vercel <vercel[bot]@users.noreply.github.com>

Author: vercel[bot]

app/layout.tsx

@@ -1,5 +1,6 @@
 import type { Metadata } from "next";
 import { Geist, Geist_Mono } from "next/font/google";
+import { Analytics } from "@vercel/analytics/next";
 import { Nav } from "@/app/_components/Nav";
 import { Footer } from "@/app/_components/Footer";
 import "./globals.css";
@@ -34,6 +35,7 @@ export default function RootLayout({
         <Nav />
         <main className="flex-1">{children}</main>
         <Footer />
+        <Analytics />
       </body>
     </html>
   );

package-lock.json

@@ -26,6 +26,7 @@
     ]
   },
   "dependencies": {
+    "@vercel/analytics": "^2.0.1",
     "next": "16.2.4",
     "react": "19.2.4",
     "react-dom": "19.2.4"