디자인: 게임천국 느낌 아이콘, 홈 단순화(큰 버튼 3개 + 탭), 헤더 로그인/계정 메뉴, 로그인 페이지 분리

- 아이콘: 게임패드 + 천사 고리 + 반짝이, scripts/pwa-icons.ts가 favicon.svg와 PNG를 한 디자인에서 생성
- 홈: 방 만들기·코드로 참여(/j)·컴퓨터와 연습 버튼, [게임 고르기 | 지금 열린 방] 탭
- 헤더: 로그인 전 [로그인] → /login, 로그인 뒤 닉네임 → [설정][로그아웃] 메뉴
- /login: 로그인만, /me(설정)는 로그인 안 했으면 /login?next=/me로

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

apps/web/public/favicon.svg

@@ -1 +1 @@
-<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 64 64"><rect width="64" height="64" rx="14" fill="#E3B76B"/><g stroke="#7a5a22" stroke-width="2"><path d="M12 22h40M12 32h40M12 42h40M22 12v40M32 12v40M42 12v40"/></g><circle cx="32" cy="32" r="8" fill="#1c1e21"/><circle cx="42" cy="22" r="8" fill="#fff" stroke="#1c1e21" stroke-width="2"/></svg>
+<svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 64 64"><defs><linearGradient id="bg" x1="0" y1="0" x2="1" y2="1"><stop offset="0" stop-color="#7c3aed"/><stop offset=".55" stop-color="#db2777"/><stop offset="1" stop-color="#f97316"/></linearGradient></defs><rect width="64" height="64" rx="14" fill="url(#bg)"/><ellipse cx="32" cy="14.5" rx="11" ry="3.6" fill="none" stroke="#fde047" stroke-width="2.6"/><path d="M20 21h24a10 10 0 0 1 9.6 7.2l3.3 11.6a5.6 5.6 0 0 1-9.5 5.3L43 40H21l-4.4 5.1a5.6 5.6 0 0 1-9.5-5.3l3.3-11.6A10 10 0 0 1 20 21z" fill="#fff" stroke="#1e1b4b" stroke-width="2.2" stroke-linejoin="round"/><path d="M19.6 26.2h2.8v3h3v2.8h-3v3h-2.8v-3h-3v-2.8h3z" fill="#1e1b4b"/><circle cx="43" cy="26.6" r="2.3" fill="#facc15" stroke="#1e1b4b" stroke-width="1"/><circle cx="46.8" cy="30.6" r="2.3" fill="#ef4444" stroke="#1e1b4b" stroke-width="1"/><circle cx="43" cy="34.6" r="2.3" fill="#22c55e" stroke="#1e1b4b" stroke-width="1"/><circle cx="39.2" cy="30.6" r="2.3" fill="#3b82f6" stroke="#1e1b4b" stroke-width="1"/><path d="M53 6.4Q53 11 57.6 11Q53 11 53 15.6Q53 11 48.4 11Q53 11 53 6.4z" fill="#fde047"/><path d="M10.5 11Q10.5 14 13.5 14Q10.5 14 10.5 17Q10.5 14 7.5 14Q10.5 14 10.5 11z" fill="#fff"/><path d="M54 50Q54 53 57 53Q54 53 54 56Q54 53 51 53Q54 53 54 50z" fill="#fff"/><path d="M11 51.8Q11 54 13.2 54Q11 54 11 56.2Q11 54 8.8 54Q11 54 11 51.8z" fill="#fde047"/></svg>