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/routeTree.gen.ts

@@ -10,6 +10,7 @@
 
 import { Route as rootRouteImport } from './routes/__root'
 import { Route as PlayRouteImport } from './routes/play'
+import { Route as LoginRouteImport } from './routes/login'
 import { Route as AboutRouteImport } from './routes/about'
 import { Route as IndexRouteImport } from './routes/index'
 
@@ -18,6 +19,11 @@ const PlayRoute = PlayRouteImport.update({
   path: '/play',
   getParentRoute: () => rootRouteImport,
 } as any)
+const LoginRoute = LoginRouteImport.update({
+  id: '/login',
+  path: '/login',
+  getParentRoute: () => rootRouteImport,
+} as any)
 const AboutRoute = AboutRouteImport.update({
   id: '/about',
   path: '/about',
@@ -32,30 +38,34 @@ const IndexRoute = IndexRouteImport.update({
 export interface FileRoutesByFullPath {
   '/': typeof IndexRoute
   '/about': typeof AboutRoute
+  '/login': typeof LoginRoute
   '/play': typeof PlayRoute
 }
 export interface FileRoutesByTo {
   '/': typeof IndexRoute
   '/about': typeof AboutRoute
+  '/login': typeof LoginRoute
   '/play': typeof PlayRoute
 }
 export interface FileRoutesById {
   __root__: typeof rootRouteImport
   '/': typeof IndexRoute
   '/about': typeof AboutRoute
+  '/login': typeof LoginRoute
   '/play': typeof PlayRoute
 }
 export interface FileRouteTypes {
   fileRoutesByFullPath: FileRoutesByFullPath
-  fullPaths: '/' | '/about' | '/play'
+  fullPaths: '/' | '/about' | '/login' | '/play'
   fileRoutesByTo: FileRoutesByTo
-  to: '/' | '/about' | '/play'
-  id: '__root__' | '/' | '/about' | '/play'
+  to: '/' | '/about' | '/login' | '/play'
+  id: '__root__' | '/' | '/about' | '/login' | '/play'
   fileRoutesById: FileRoutesById
 }
 export interface RootRouteChildren {
   IndexRoute: typeof IndexRoute
   AboutRoute: typeof AboutRoute
+  LoginRoute: typeof LoginRoute
   PlayRoute: typeof PlayRoute
 }
 
@@ -68,6 +78,13 @@ declare module '@tanstack/react-router' {
       preLoaderRoute: typeof PlayRouteImport
       parentRoute: typeof rootRouteImport
     }
+    '/login': {
+      id: '/login'
+      path: '/login'
+      fullPath: '/login'
+      preLoaderRoute: typeof LoginRouteImport
+      parentRoute: typeof rootRouteImport
+    }
     '/about': {
       id: '/about'
       path: '/about'
@@ -88,6 +105,7 @@ declare module '@tanstack/react-router' {
 const rootRouteChildren: RootRouteChildren = {
   IndexRoute: IndexRoute,
   AboutRoute: AboutRoute,
+  LoginRoute: LoginRoute,
   PlayRoute: PlayRoute,
 }
 export const routeTree = rootRouteImport