Add Authentication With Email and Google

Add authentication to this project.

Include:
- An auth page with sign-up and sign-in using email and password, plus Google sign-in.
- Sign-out available from the header when signed in.
- Session persistence across refreshes.
- A protected area that signed-out visitors are redirected away from.
- A profile record created automatically for every new user.

Important:
- The social sign-in redirect must point at a public URL, not directly at a protected page.
- Protect the data as well as the pages — signed-out requests must not return private rows.
- Handle the "already registered" and "wrong password" cases with readable messages.

Acceptance criteria: I can sign up, sign out, sign back in, refresh, and still be signed in.