A beautiful Endgame Study

I have recently tried to solve this study, but was not able to find the correct solution or even find the right path to the solution. The solution turned out to be so beautiful that I had to share the study and also annotate to solution. So I hope you will enjoy it as much as I did.

Black's a-pawn is unstoppable and white has no direct way to mate the king. The h-pawn is also slower than the black a-pawn, so the situation seems hopeless for white.
1.Kg6
White makes way for the h-pawn, but isn't this too slow?
1...a3 2.h6 a2 3.h7 a1=Q 4.h8=Q!
White promotes their pawn anyway, even though it can be taken.
4...Qxh8 5.f6! This is white's brilliant idea, the black queen is now trapped in the corner. White cannot win the queen because it cannot be attacked, but white will be able to force Zugzwang.
5...h3
Black tries to get the queen out, so white's answer is forced.
6.Kg5 d4
White has an important decision to make.
7.c6!
Not 7.exd4? d5! 8.cxd6 Qh4+ 9.Kxh4 stalemate.
7...dxc6 8.exd4 c5 9.d5!
Again, white cannot capture the pawn because of the stalemate threat. White has correctly navigated every obstacle and the last moves are obvious.
9...c4 10.d6 c3 11.d7 c2 12.d8=Q#