feat(api): add auth middleware to protect game endpoints

- Add requireAuth middleware using Better Auth session validation
- Apply to all game routes (start, answer)
- Unauthenticated requests return 401

apps/web/src/routes/login.tsx

@@ -0,0 +1,44 @@
+import { createFileRoute, useNavigate } from "@tanstack/react-router";
+import { signIn, useSession } from "../lib/auth-client";
+
+const LoginPage = () => {
+  const { data: session, isPending } = useSession();
+  const navigate = useNavigate();
+
+  if (isPending) return <div className="p-4">Loading...</div>;
+
+  if (session) {
+    navigate({ to: "/" });
+    return null;
+  }
+
+  return (
+    <div className="flex flex-col items-center justify-center gap-4 p-8">
+      <h1 className="text-2xl font-bold">Sign in to Glossa</h1>
+      <button
+        className="w-64 rounded bg-gray-800 px-4 py-2 text-white hover:bg-gray-700"
+        onClick={() =>
+          signIn.social({
+            provider: "github",
+            callbackURL: "http://localhost:5173",
+          })
+        }
+      >
+        Continue with GitHub
+      </button>
+      <button
+        className="w-64 rounded bg-blue-600 px-4 py-2 text-white hover:bg-blue-500"
+        onClick={() =>
+          signIn.social({
+            provider: "google",
+            callbackURL: "http://localhost:5173",
+          })
+        }
+      >
+        Continue with Google
+      </button>
+    </div>
+  );
+};
+
+export const Route = createFileRoute("/login")({ component: LoginPage });