implement better-auth auth with postgres and route protection

package.json

@@ -10,16 +10,19 @@
   },
   "dependencies": {
     "@tailwindcss/postcss": "^4.2.1",
+    "better-auth": "^1.4.19",
     "clsx": "^2.1.1",
     "date-fns": "^4.1.0",
     "lucide-react": "^0.575.0",
     "next": "^16.1.6",
+    "pg": "^8.18.0",
     "react": "^19.2.4",
     "react-dom": "^19.2.4",
     "recharts": "^3.7.0"
   },
   "devDependencies": {
     "@types/node": "^25.3.0",
+    "@types/pg": "^8.16.0",
     "@types/react": "^19.2.14",
     "@types/react-dom": "^19.2.3",
     "autoprefixer": "^10.4.24",