Are The Black Pawns Trouble?

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The given position is an endgame study by Archakov & Zinar, published in Shakhmaty v SSSR in 1986, with white to play and win by its king keeping its distance from the would-be deflecting and then troubling pawns:

1. Kg1 to enter the square of the queenside passers without deflection as 1. Kg3 h4+ and 1. Kg2 h4 2. a6 h3+ both lose. The a pawn will stop the h pawn.

if 1... d3 2. Kf1 (not 2. Kf2 g5 3. a6 g4 4. a7 g3+) 2... c4 3. a6 and white wins

if 1... c4 2. Kf1 c3 3. Ke1 and white wins (not 3. Ke2 f5 4. a6 f4 5. a7 f3+ 6. Kxf3 c2)

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