docs: improve Fumadocs search and AI integration

apps/docs/package.json

@@ -14,7 +14,8 @@
     "clean": "git clean -xdf .next out node_modules",
     "api-dump-contributors": "node ./scripts/dump-contributors.mjs",
     "validate:coverage": "tsx ./scripts/validate-docs-coverage.ts",
-    "validate:links": "tsx ./scripts/validate-links.ts"
+    "validate:links": "tsx ./scripts/validate-links.ts",
+    "verify:search": "tsx ./scripts/verify-search.ts"
   },
   "dependencies": {
     "@base-ui/react": "^1.5.0",
@@ -25,20 +26,17 @@
     "@mdx-js/react": "^3.1.1",
     "@nivo/line": "^0.99.0",
     "@scalar/api-client-react": "catalog:",
-    "@shadcn/react": "catalog:",
     "@tabler/icons-react": "catalog:",
     "@tailwindcss/postcss": "^4.3.2",
     "class-variance-authority": "^0.7.1",
     "clsx": "catalog:",
     "date-fns": "catalog:",
     "dayjs": "catalog:",
-    "embla-carousel-react": "catalog:",
     "feed": "catalog:",
     "fumadocs-core": "catalog:",
     "fumadocs-mdx": "catalog:",
     "fumadocs-openapi": "catalog:",
     "fumadocs-ui": "catalog:",
-    "input-otp": "catalog:",
     "json-schema-typed": "catalog:",
     "mermaid": "catalog:",
     "next": "catalog:",
@@ -48,8 +46,6 @@
     "posthog-js": "catalog:",
     "react": "catalog:",
     "react-dom": "catalog:",
-    "react-resizable-panels": "catalog:",
-    "recharts": "3.8.0",
     "remark-directive": "catalog:",
     "shiki": "catalog:",
     "sonner": "catalog:",