So I assumed the "without I/O" clue was a hint to Base58Check encoding. Given that, if you take "BTCETHXRPPhemex" as a Base58Check string and convert it to Base10, you end up with a 27 digit number.

I took every combination of the words "BTC", "ETH", "XRP", and "Phemex" (24 combinations in total), and converted from Base58Check to Base10.

For every one of these 27-digit numbers (24 in total), I concatenated them before the prime, concatenated after the prime, and multiplied the two together.

For each of these 72 resulting numbers, I tried:
Hashing them with SHA256
Hashing them twice with SHA256 (the "Go back to step 4 again" clue makes me think we have to do something twice)
Converting them to HEX, then hashing them

I used each of the 216 results as a private key and checked the addresses. No luck.

I'm out of ideas at this point. Going to have to wait for another hint.