feat(api): add WebSocket foundation and multiplayer game store

- Add ws/ directory: server setup, auth, router, connections map
- WebSocket auth rejects upgrade with 401 if no Better Auth session
- Router parses WsClientMessageSchema, dispatches to handlers,
  two-layer error handling (AppError -> WsErrorSchema, unknown -> 500)
- connections.ts: in-memory Map<lobbyId, Map<userId, WebSocket>>
  with addConnection, removeConnection, broadcastToLobby
- LobbyGameStore interface + InMemoryLobbyGameStore implementation
  following existing GameSessionStore pattern
- multiplayerGameService: generateMultiplayerQuestions() decoupled
  from single-player flow, hardcoded defaults en->it nouns easy 3 rounds
- handleLobbyJoin and handleLobbyLeave implemented
- WsErrorSchema added to shared schemas
- server.ts switched to createServer + setupWebSocket

apps/api/src/services/multiplayerGameService.ts

@@ -0,0 +1,75 @@
+import { randomUUID } from "crypto";
+import { getGameTerms, getDistractors } from "@lila/db";
+import type {
+  GameQuestion,
+  AnswerOption,
+  SupportedLanguageCode,
+  SupportedPos,
+  DifficultyLevel,
+} from "@lila/shared";
+
+// TODO(game-mode-slice): replace with lobby settings when mode selection lands
+const MULTIPLAYER_DEFAULTS = {
+  sourceLanguage: "en" as SupportedLanguageCode,
+  targetLanguage: "it" as SupportedLanguageCode,
+  pos: "noun" as SupportedPos,
+  difficulty: "easy" as DifficultyLevel,
+  rounds: 3,
+};
+
+const shuffle = <T>(array: T[]): T[] => {
+  const result = [...array];
+  for (let i = result.length - 1; i > 0; i--) {
+    const j = Math.floor(Math.random() * (i + 1));
+    const temp = result[i]!;
+    result[i] = result[j]!;
+    result[j] = temp;
+  }
+  return result;
+};
+
+export type MultiplayerQuestion = GameQuestion & { correctOptionId: number };
+
+export const generateMultiplayerQuestions = async (): Promise<
+  MultiplayerQuestion[]
+> => {
+  const correctAnswers = await getGameTerms(
+    MULTIPLAYER_DEFAULTS.sourceLanguage,
+    MULTIPLAYER_DEFAULTS.targetLanguage,
+    MULTIPLAYER_DEFAULTS.pos,
+    MULTIPLAYER_DEFAULTS.difficulty,
+    MULTIPLAYER_DEFAULTS.rounds,
+  );
+
+  const questions: MultiplayerQuestion[] = await Promise.all(
+    correctAnswers.map(async (correctAnswer) => {
+      const distractorTexts = await getDistractors(
+        correctAnswer.termId,
+        correctAnswer.targetText,
+        MULTIPLAYER_DEFAULTS.targetLanguage,
+        MULTIPLAYER_DEFAULTS.pos,
+        MULTIPLAYER_DEFAULTS.difficulty,
+        3,
+      );
+
+      const optionTexts = [correctAnswer.targetText, ...distractorTexts];
+      const shuffledTexts = shuffle(optionTexts);
+      const correctOptionId = shuffledTexts.indexOf(correctAnswer.targetText);
+
+      const options: AnswerOption[] = shuffledTexts.map((text, index) => ({
+        optionId: index,
+        text,
+      }));
+
+      return {
+        questionId: randomUUID(),
+        prompt: correctAnswer.sourceText,
+        gloss: correctAnswer.sourceGloss,
+        options,
+        correctOptionId,
+      };
+    }),
+  );
+
+  return questions;
+};