I've created a complete authentication system with login and signup functionality that includes social login options for Google, GitHub, and LinkedIn. The system includes:

1. A login page with:

   • Email/password login form
   • Social login buttons (Google, GitHub, LinkedIn)
   • "Forgot password" link
   • Link to sign up for new users

2. A signup page with:

   • Registration form (name, email, password)
   • Same social login options
   • Link back to the login page
3. A dashboard page that displays user information after successful login
4. Server-side authentication logic using Next.js Server Actions and API routes

The authentication flow works as follows:

• Users can sign in with email/password or choose a social login option
• For email/password login, credentials are validated server-side
• For social logins, the user is redirected to the provider's authentication page
• After successful authentication, users are redirected to the dashboard

This implementation uses cookies for session management and includes proper error handling. In a production environment, you would want to:

1. Use a proper authentication library like NextAuth.js or Clerk
2. Store user data in a database rather than in-memory
3. Implement proper password hashing
4. Set up proper OAuth flows for the social login providers

You can deploy this to Vercel and it will work out of the box for demonstration purposes. For a production application, you would need to set up the proper OAuth credentials with each provider.