feat(api): add answer evaluation endpoint

Complete the game answer flow:

- Add evaluateAnswer service function: looks up the session in the
  GameSessionStore, compares the submitted optionId against the stored
  correct answer, returns an AnswerResult.
- Add submitAnswer controller with safeParse validation and error
  handling (session/question not found → 404).
- Add POST /api/v1/game/answer route.
- Fix createGameSession: was missing the answerKey tracking and the
  gameSessionStore.create() call, so sessions were never persisted.

The full singleplayer game loop now works end-to-end:
POST /game/start → GameSession, POST /game/answer → AnswerResult.

scripts/gametest/test-game.ts

@@ -1,5 +1,6 @@
 async function main() {
-  const response = await fetch("http://localhost:3000/api/v1/game/start", {
+  // Step 1: start a game
+  const startResponse = await fetch("http://localhost:3000/api/v1/game/start", {
     method: "POST",
     headers: { "Content-Type": "application/json" },
     body: JSON.stringify({
@@ -10,9 +11,29 @@ async function main() {
       rounds: "3",
     }),
   });
+  const game = await startResponse.json();
+  console.log("Game started:", JSON.stringify(game, null, 2));
 
-  const data = await response.json();
-  console.log(JSON.stringify(data, null, 2));
+  // Step 2: answer each question (always pick option 0)
+  for (const question of game.data.questions) {
+    const answerResponse = await fetch(
+      "http://localhost:3000/api/v1/game/answer",
+      {
+        method: "POST",
+        headers: { "Content-Type": "application/json" },
+        body: JSON.stringify({
+          sessionId: game.data.sessionId,
+          questionId: question.questionId,
+          selectedOptionId: 0,
+        }),
+      },
+    );
+    const result = await answerResponse.json();
+    console.log("Raw result:", JSON.stringify(result, null, 2));
+    console.log(
+      `${question.prompt}: ${result.data.isCorrect ? "✓" : "✗"} (picked ${0}, correct was ${result.data.correctOptionId})`,
+    );
+  }
 }
 
 main();