Stuck in a loop when signing in

I'm trying to sign on to my icloud account. It lets me enter my name and password. Then it wants to send a six digit verification code to my rescue email address. I do that, enter the verification code, and then it says I have to verify my @mac.com email address, which is the one I signed in with. I click submit and it says no you can't use that email address. I try the same name but with @me.com and it again says no you can't use that email address. I try the rescue email address and it tells me "you can't use your rescue email address." What is going on here? I give it yet another email address and then it seems to want me to use that address to create an icloud ID. But I have an icloud id. That's what I signed on with in the beginning. What is going on here? I seem to be stuck in some kind of loop. Why is it making me verify after I already signed in with account and password and verification code?