Putting a domain on my pages.dev site

Soooo I have successfully made and launched my website using pages, which is deployed from a GitHub repository. The website is fully functional, and I am now ready to acquire a custom domain for it

My understanding is that I can register the desired domain directly through cloudflare's domain registration? Since Cloudflare is already hosting the site as i understand, I am looking for clarification on the other necessary steps

Specifically, after purchasing the domain will it automatically be configured to point to my existing pages site, or are there additional configuration steps I need to doo? I would appreciate help on the complete process to ensure a smooth transition to a custom domain.
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