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

vitest.config.mts

@@ -0,0 +1,11 @@
+import { defineConfig } from "vitest/config"
+import react from "@vitejs/plugin-react"
+import tsconfigPaths from "vite-tsconfig-paths"
+
+export default defineConfig({
+    plugins: [tsconfigPaths(), react()],
+    test: {
+        environment: "jsdom",
+        include: ["tests/**/*.test.{ts,tsx}"],
+    },
+})