test: add Vitest and Playwright testing infrastructure

- Add Vitest for unit tests (39 tests)
  - cached-responses.test.ts
  - ai-providers.test.ts
  - chat-helpers.test.ts
  - utils.test.ts
- Add Playwright for E2E tests (3 smoke tests)
  - Homepage load
  - Japanese locale
  - Settings dialog
- Add CI workflow (.github/workflows/test.yml)
- Add vitest.config.mts and playwright.config.ts
- Update .gitignore for test artifacts

.gitignore

@@ -14,6 +14,8 @@ packages/*/dist
 
 # testing
 /coverage
+/playwright-report/
+/test-results/
 
 # next.js
 /.next/