refactor: migrate authentication system and update Drizzle schema.

src/app/api/auth/[...all]/route.ts

@@ -0,0 +1,4 @@
+import { toNextJsHandler } from "better-auth/next-js";
+import { auth } from "~/lib/auth";
+
+export const { GET, POST } = toNextJsHandler(auth);

src/app/api/auth/[...nextauth]/route.ts

@@ -1,3 +0,0 @@
-import { handlers } from "~/server/auth";
-
-export const { GET, POST } = handlers;