Ok, so 1.Kg2? e5! probably draws. That means 1.f4 is the only way to win. Nice. Was this in Van Perlo's Endgame Tactics? It feels familiar.
Tricky pawn endgame
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My first choice would be f4 here. b3-b4 is a waste of a tempo. In K+P endings, you should save those sorts of pawn moves until they're necessary.
But I'd be surprised if 1.Kg2 didn't win.