feat(web): add multiplayer layout route and landing page

- multiplayer.tsx: layout route wrapping all multiplayer children
  with WsProvider, auth guard via beforeLoad
- multiplayer/index.tsx: create/join landing page
  - POST /api/v1/lobbies to create, navigates to lobby waiting room
  - POST /api/v1/lobbies/:code/join to join, normalizes code to
    uppercase before sending
  - loading states per action, error display, Enter key on join input
  - imports Lobby type from @lila/shared (single source of truth)

apps/web/src/routeTree.gen.ts

@@ -10,15 +10,24 @@
 
 import { Route as rootRouteImport } from './routes/__root'
 import { Route as PlayRouteImport } from './routes/play'
+import { Route as MultiplayerRouteImport } from './routes/multiplayer'
 import { Route as LoginRouteImport } from './routes/login'
 import { Route as AboutRouteImport } from './routes/about'
 import { Route as IndexRouteImport } from './routes/index'
+import { Route as MultiplayerIndexRouteImport } from './routes/multiplayer/index'
+import { Route as MultiplayerLobbyCodeRouteImport } from './routes/multiplayer/lobby.$code'
+import { Route as MultiplayerGameCodeRouteImport } from './routes/multiplayer/game.$code'
 
 const PlayRoute = PlayRouteImport.update({
   id: '/play',
   path: '/play',
   getParentRoute: () => rootRouteImport,
 } as any)
+const MultiplayerRoute = MultiplayerRouteImport.update({
+  id: '/multiplayer',
+  path: '/multiplayer',
+  getParentRoute: () => rootRouteImport,
+} as any)
 const LoginRoute = LoginRouteImport.update({
   id: '/login',
   path: '/login',
@@ -34,38 +43,89 @@ const IndexRoute = IndexRouteImport.update({
   path: '/',
   getParentRoute: () => rootRouteImport,
 } as any)
+const MultiplayerIndexRoute = MultiplayerIndexRouteImport.update({
+  id: '/',
+  path: '/',
+  getParentRoute: () => MultiplayerRoute,
+} as any)
+const MultiplayerLobbyCodeRoute = MultiplayerLobbyCodeRouteImport.update({
+  id: '/lobby/$code',
+  path: '/lobby/$code',
+  getParentRoute: () => MultiplayerRoute,
+} as any)
+const MultiplayerGameCodeRoute = MultiplayerGameCodeRouteImport.update({
+  id: '/game/$code',
+  path: '/game/$code',
+  getParentRoute: () => MultiplayerRoute,
+} as any)
 
 export interface FileRoutesByFullPath {
   '/': typeof IndexRoute
   '/about': typeof AboutRoute
   '/login': typeof LoginRoute
+  '/multiplayer': typeof MultiplayerRouteWithChildren
   '/play': typeof PlayRoute
+  '/multiplayer/': typeof MultiplayerIndexRoute
+  '/multiplayer/game/$code': typeof MultiplayerGameCodeRoute
+  '/multiplayer/lobby/$code': typeof MultiplayerLobbyCodeRoute
 }
 export interface FileRoutesByTo {
   '/': typeof IndexRoute
   '/about': typeof AboutRoute
   '/login': typeof LoginRoute
   '/play': typeof PlayRoute
+  '/multiplayer': typeof MultiplayerIndexRoute
+  '/multiplayer/game/$code': typeof MultiplayerGameCodeRoute
+  '/multiplayer/lobby/$code': typeof MultiplayerLobbyCodeRoute
 }
 export interface FileRoutesById {
   __root__: typeof rootRouteImport
   '/': typeof IndexRoute
   '/about': typeof AboutRoute
   '/login': typeof LoginRoute
+  '/multiplayer': typeof MultiplayerRouteWithChildren
   '/play': typeof PlayRoute
+  '/multiplayer/': typeof MultiplayerIndexRoute
+  '/multiplayer/game/$code': typeof MultiplayerGameCodeRoute
+  '/multiplayer/lobby/$code': typeof MultiplayerLobbyCodeRoute
 }
 export interface FileRouteTypes {
   fileRoutesByFullPath: FileRoutesByFullPath
-  fullPaths: '/' | '/about' | '/login' | '/play'
+  fullPaths:
+    | '/'
+    | '/about'
+    | '/login'
+    | '/multiplayer'
+    | '/play'
+    | '/multiplayer/'
+    | '/multiplayer/game/$code'
+    | '/multiplayer/lobby/$code'
   fileRoutesByTo: FileRoutesByTo
-  to: '/' | '/about' | '/login' | '/play'
-  id: '__root__' | '/' | '/about' | '/login' | '/play'
+  to:
+    | '/'
+    | '/about'
+    | '/login'
+    | '/play'
+    | '/multiplayer'
+    | '/multiplayer/game/$code'
+    | '/multiplayer/lobby/$code'
+  id:
+    | '__root__'
+    | '/'
+    | '/about'
+    | '/login'
+    | '/multiplayer'
+    | '/play'
+    | '/multiplayer/'
+    | '/multiplayer/game/$code'
+    | '/multiplayer/lobby/$code'
   fileRoutesById: FileRoutesById
 }
 export interface RootRouteChildren {
   IndexRoute: typeof IndexRoute
   AboutRoute: typeof AboutRoute
   LoginRoute: typeof LoginRoute
+  MultiplayerRoute: typeof MultiplayerRouteWithChildren
   PlayRoute: typeof PlayRoute
 }
 
@@ -78,6 +138,13 @@ declare module '@tanstack/react-router' {
       preLoaderRoute: typeof PlayRouteImport
       parentRoute: typeof rootRouteImport
     }
+    '/multiplayer': {
+      id: '/multiplayer'
+      path: '/multiplayer'
+      fullPath: '/multiplayer'
+      preLoaderRoute: typeof MultiplayerRouteImport
+      parentRoute: typeof rootRouteImport
+    }
     '/login': {
       id: '/login'
       path: '/login'
@@ -99,13 +166,51 @@ declare module '@tanstack/react-router' {
       preLoaderRoute: typeof IndexRouteImport
       parentRoute: typeof rootRouteImport
     }
+    '/multiplayer/': {
+      id: '/multiplayer/'
+      path: '/'
+      fullPath: '/multiplayer/'
+      preLoaderRoute: typeof MultiplayerIndexRouteImport
+      parentRoute: typeof MultiplayerRoute
+    }
+    '/multiplayer/lobby/$code': {
+      id: '/multiplayer/lobby/$code'
+      path: '/lobby/$code'
+      fullPath: '/multiplayer/lobby/$code'
+      preLoaderRoute: typeof MultiplayerLobbyCodeRouteImport
+      parentRoute: typeof MultiplayerRoute
+    }
+    '/multiplayer/game/$code': {
+      id: '/multiplayer/game/$code'
+      path: '/game/$code'
+      fullPath: '/multiplayer/game/$code'
+      preLoaderRoute: typeof MultiplayerGameCodeRouteImport
+      parentRoute: typeof MultiplayerRoute
+    }
   }
 }
 
+interface MultiplayerRouteChildren {
+  MultiplayerIndexRoute: typeof MultiplayerIndexRoute
+  MultiplayerGameCodeRoute: typeof MultiplayerGameCodeRoute
+  MultiplayerLobbyCodeRoute: typeof MultiplayerLobbyCodeRoute
+}
+
+const MultiplayerRouteChildren: MultiplayerRouteChildren = {
+  MultiplayerIndexRoute: MultiplayerIndexRoute,
+  MultiplayerGameCodeRoute: MultiplayerGameCodeRoute,
+  MultiplayerLobbyCodeRoute: MultiplayerLobbyCodeRoute,
+}
+
+const MultiplayerRouteWithChildren = MultiplayerRoute._addFileChildren(
+  MultiplayerRouteChildren,
+)
+
 const rootRouteChildren: RootRouteChildren = {
   IndexRoute: IndexRoute,
   AboutRoute: AboutRoute,
   LoginRoute: LoginRoute,
+  MultiplayerRoute: MultiplayerRouteWithChildren,
   PlayRoute: PlayRoute,
 }
 export const routeTree = rootRouteImport