To publish with a custom domain name, follow these steps:

1. Publish first - Click the "Publish" button in the top right of the chat to deploy your project to Vercel. This gives you a default .vercel.app URL.
2. Add a custom domain - After publishing, go to your project on vercel.com/dashboard, select your project, then navigate to Settings > Domains. There you can add your custom domain (e.g. merlinai.com).
3. Update DNS records - Vercel will give you DNS records (usually a CNAME or A record) that you need to add with your domain registrar (wherever you bought the domain, like Namecheap, GoDaddy, Google Domains, etc.).
4. Verification - Once the DNS records propagate (can take a few minutes to 48 hours), Vercel will automatically verify the domain and set up HTTPS for you.

If you don't have a domain yet, you can purchase one through Vercel directly at Settings > Domains > Buy, or from any domain registrar.

It depends on where you are in your process:

• Preview - Use this while you're still testing and making changes. Preview deployments are generated for every push/update and are not your live public site. Good for checking things work before going live.
• Production - Use this when you're ready for the public-facing live version of your app. This is the deployment tied to your custom domain.

Recommendation: Select both Preview and Production when adding environment variables (like your Groq API key). That way your app works correctly in both environments - while you're testing changes and on the live site. Most environment variable forms in Vercel let you tick multiple environments at once, so there's no downside to enabling both.