feat(web): add minimal playable quiz at /play

- Add Vite proxy for /api → localhost:3000 (no CORS needed in dev)
- Create /play route with hardcoded game settings (en→it, nouns, easy)
- Three-phase state machine: loading → playing → finished
- Show prompt, optional gloss, and 4 answer buttons per question
- Submit answers to /api/v1/game/answer, show correct/wrong feedback
- Manual Next button to advance after answering
- Score screen on completion
- Add selectedOptionId to AnswerResult schema (discovered during
  frontend work that the result needs to be self-contained for
  rendering feedback without separate client state)

Intentionally unstyled — component extraction and polish come next.

apps/web/src/routeTree.gen.ts

@@ -9,9 +9,15 @@
 // Additionally, you should also exclude this file from your linter and/or formatter to prevent it from being checked or modified.
 
 import { Route as rootRouteImport } from './routes/__root'
+import { Route as PlayRouteImport } from './routes/play'
 import { Route as AboutRouteImport } from './routes/about'
 import { Route as IndexRouteImport } from './routes/index'
 
+const PlayRoute = PlayRouteImport.update({
+  id: '/play',
+  path: '/play',
+  getParentRoute: () => rootRouteImport,
+} as any)
 const AboutRoute = AboutRouteImport.update({
   id: '/about',
   path: '/about',
@@ -26,31 +32,42 @@ const IndexRoute = IndexRouteImport.update({
 export interface FileRoutesByFullPath {
   '/': typeof IndexRoute
   '/about': typeof AboutRoute
+  '/play': typeof PlayRoute
 }
 export interface FileRoutesByTo {
   '/': typeof IndexRoute
   '/about': typeof AboutRoute
+  '/play': typeof PlayRoute
 }
 export interface FileRoutesById {
   __root__: typeof rootRouteImport
   '/': typeof IndexRoute
   '/about': typeof AboutRoute
+  '/play': typeof PlayRoute
 }
 export interface FileRouteTypes {
   fileRoutesByFullPath: FileRoutesByFullPath
-  fullPaths: '/' | '/about'
+  fullPaths: '/' | '/about' | '/play'
   fileRoutesByTo: FileRoutesByTo
-  to: '/' | '/about'
-  id: '__root__' | '/' | '/about'
+  to: '/' | '/about' | '/play'
+  id: '__root__' | '/' | '/about' | '/play'
   fileRoutesById: FileRoutesById
 }
 export interface RootRouteChildren {
   IndexRoute: typeof IndexRoute
   AboutRoute: typeof AboutRoute
+  PlayRoute: typeof PlayRoute
 }
 
 declare module '@tanstack/react-router' {
   interface FileRoutesByPath {
+    '/play': {
+      id: '/play'
+      path: '/play'
+      fullPath: '/play'
+      preLoaderRoute: typeof PlayRouteImport
+      parentRoute: typeof rootRouteImport
+    }
     '/about': {
       id: '/about'
       path: '/about'
@@ -71,6 +88,7 @@ declare module '@tanstack/react-router' {
 const rootRouteChildren: RootRouteChildren = {
   IndexRoute: IndexRoute,
   AboutRoute: AboutRoute,
+  PlayRoute: PlayRoute,
 }
 export const routeTree = rootRouteImport
   ._addFileChildren(rootRouteChildren)