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The key here is to not let white get the opposition with his king on the 6th rank. For white to win, he needs to be a square directly in front of the pawn with the opposition, or two without.
If the white king is on the same rank as the pawn and black can prevent white from getting into the above positions, it's a draw.
Yeah, I just went through it again myself. I realized that I never fully considered the simple option of the black king retreating, as opposed to moving side to side. That would explain how I missed it.
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