Custom API Gateway Stage Names

I am trying to figure out how to have a different name for my API gateway’s stages. For example, we want our production environment to use the stage ‘prod’ when configuring and deploying but our CloudFront expect the api gateway’s stage to be named ‘api’. I found an old topic but it had no working solution. Am I missing something simple here?

For each of your deployment stages (I’m guessing you only have one or two) you can create a “custom domain name” in the API gateway console.
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