Add GitHub App token retrieval step to workflow

Author: BigSpaceships

.github/workflows/sync-branches.yml

@@ -15,8 +15,15 @@ jobs:
 
     steps:
       - uses: actions/checkout@v5
+      - name: Get PAT for app
+        uses: actions/create-github-app-token@v2
+        id: app-token
+        with:
+          app-id: ${{ secrets.APP_ID }}
+          private-key: ${{ secrets.APP_SECRET }}
       - name: Push changes
         uses: ad-m/github-push-action@master
         with:
+          github_token: ${{ steps.app-token.outputs.token }}
           branch: main
           force: true