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The only first move that wins is 1.Kd5
Yeah, I know Kd5 is the first move. g5 draws.
I think I got it
1.g5 is stupid- Black will just move the king back and forth without altering the pawn structure.
White has to plant a pawn at f6 to win, and Black can't prevent this (an eventual 1...f5 is answered by g4-g5).
So, 1,Kd5 is the only serious move, and Black apparently can't defend to the simple plan of bringing the king to e7/e8, and pushing f4-f5. There is only one detail: after a pawn swap at f5 white must have the opposition- but this is rather easy to arrange, the corresponding squares are e8/g7 and e7/g8.
I am glad to have an IM explain the position in detail. This all makes sense! I will try it out soon. Thank you, Mr. Frendzas!
He also has four stupid continuations: 3...Kd7/d8/e8/f8 - and white cannot make any progress.