fix: fix typos and usage errors of next auth (#778)

Co-authored-by: imWildCat <[email protected]>

docs/quick-starts/framework/next-auth/_config-provider.mdx

@@ -18,11 +18,11 @@ import SignInFlowSummary from '../../fragments/_web-sign-in-flow-summary.mdx';
   redirectUri="http://localhost:3000/api/auth/callback/logto"
 />
 
-#### Config Next Auth provider
+#### Set up Next Auth provider
 
 <GetAppSecret />
 
-Modify your API route config of Next Auth, if you are using Pages Router, the file is in `pages/api/auth/[...nextauth].js`, if you are using App Router, the file is in `app/api/auth/[...nextauth]/router.ts`.
+Modify your API route config of Next Auth, if you are using Pages Router, the file is in `pages/api/auth/[...nextauth].js`, if you are using App Router, the file is in `app/api/auth/[...nextauth]/route.ts`.
 
 The following is an example of App Router:
 
@@ -33,7 +33,12 @@ The following is an example of App Router:
 ```ts
 import NextAuth from 'next-auth';
 
-export const { handlers, signIn, signOut, auth } = NextAuth({
+export const {
+  handlers: { GET, POST },
+  signIn,
+  signOut,
+  auth,
+} = NextAuth({
   providers: [
     {
       id: 'logto',