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Can Someone Explain this puzzle?

If you want to be even more precise, white will always have at least one extra tempo thanks to the pawns on a3 and b2 (if black plays b5 white can just delay and play c5 later to break through on two sides of the board). This will come in handy if black tries to take the opposition. White will then be able to enter black's position through either d6 or f6, and slowly but surely push the g- and h-pawns :)
lichess.org/training/F0owi
The point is that the white king wants to get the centre so there will be no way that black can hold the game he will have some pawn moves but we have too, so after some moves black is gona run out of pawn moves and it wil be very hard for black to hold the game and he will lose it after black is forced to move the king away. And give the d6 or the f6 squares and the white king will enter one of those squares and take the pawns and create a past pawn and eventually win the game.
You all explained why white king should infiltrate black’s defense but noone explained why only fxg5 works. The point of making fxg5 and not hxg5 is to keep the threat of h5! that would create a passed pawn. Thus the black king can’t go too far away.

This is critical in the line : fxg5 30.hxg5 Ke7 31.Ke5 b5 32.b3 b4 33.axb4 cxb4 34.c5
At the end of this sequence, black cannot stop the c5 pawn without conceding h5. If hxg5 was played, black could stop the c pawn and draw the game.

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