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

playwright.config.ts

@@ -0,0 +1,30 @@
+import { defineConfig } from "@playwright/test"
+
+export default defineConfig({
+    testDir: "./tests/e2e",
+    fullyParallel: true,
+    forbidOnly: !!process.env.CI,
+    retries: process.env.CI ? 2 : 0,
+    workers: process.env.CI ? 1 : undefined,
+    reporter: "html",
+    webServer: {
+        command: process.env.CI
+            ? "npm run build && npm run start"
+            : "npm run dev",
+        port: process.env.CI ? 6001 : 6002,
+        reuseExistingServer: !process.env.CI,
+        timeout: 120 * 1000,
+    },
+    use: {
+        baseURL: process.env.CI
+            ? "http://localhost:6001"
+            : "http://localhost:6002",
+        trace: "on-first-retry",
+    },
+    projects: [
+        {
+            name: "chromium",
+            use: { browserName: "chromium" },
+        },
+    ],
+})