How to create vercel alias using Vercel REST API

To enhance my workflow, I’m transitioning from using the Vercel CLI command vercel alias to generate aliases, to leveraging the Vercel REST API. The API documentation specifies using the deploymentID parameter to assign an alias (https://vercel.com/docs/rest-api/endpoints/aliases#assign-an-alias).

Here’s a snippet of my current workflow:

 - name: Build and deploy
    id: build-and-deploy
    run: |
      vercel build --debug --token=${{ secrets.VERCEL_TOKEN }} --scope=***
      DEPLOYMENT_URL=$(vercel deploy --prebuilt --yes --token=${{ secrets.VERCEL_TOKEN }} --scope=***)
      echo "DEPLOYMENT_URL=${DEPLOYMENT_URL}" >> "$GITHUB_OUTPUT"

  - name: Assign aliases
    run: |
        SHORT_SHA=`echo ${GITHUB_SHA} | cut -c1-8`
        vercel alias ${{ steps.build-and-deploy.outputs.DEPLOYMENT_URL }} ${SHORT_SHA}.example.com --token=${{ secrets.VERCEL_TOKEN }} --scope=*** | awk '{print $3}')" >> "$GITHUB_OUTPUT"

How can I obtain the deployment ID and integrate it into the Vercel REST API call for alias assignment? Any insights or suggestions would be greatly appreciated.