Calculate Triangulations Fast

I was studying endgames and came across this position. It seems to require a lot of time to check lots of moves, maybe around 10-20 minutes. I wanted a method to reliably solve it in 10-20 seconds. So after lots of analysis and thinking this is what I came up with. If you have a faster method, or if this method has flaws, please let me know.

We need to solve two problems:

Problem 1: d2-e2 squares are useless, black just shuffles on c6-d6. 

Solution 1: The white king has to journey to the kingside and attack the e4 pawn from f4 (the critical square).

Problem 2: Both White and Black have passers, both at 3 squares from promotion. The pawns might race and promote at the same time. Therefore, the Black king in NOT necessarily tied down to the a5-d5-d8 square; it’s tied ONLY if the White king can stop the e pawn. 

Solution 2: The White king has to remain on the second rank OR be on the third rank, with the black king on d5, with black to move OR on the fourth rank with black king on the 6th rank, with black to move.

Now that we understand the position, let’s chart the course backwards, then forward. 

Backwards: Since the critical square is f4, we have to look at the squares that give access to it, so g3 and g4 (again critical squares). The only square that gives access to both is h3

Forward: Kf2-Kg2(avoid g3 because is critical), next move is Kh3 = Black will move twice.

Can black threaten Kd4? No, we have not yet left the second rank. Kd5? It corresponds to the g3 square, and we have access during our journey.

Conclusion: It’s winning!

1. Kf4 Kd4 is a draw, both players get queens. The puzzle is slightly more complicated than you think it is.

1 Kf2 wins.
The method is that of corresponding squares.
https://en.wikipedia.org/wiki/Corresponding_squares 

Isn't this just corresponding squares? First, find the key squares where if your king steps on it you will win no matter whose move it is (by understanding what your plan is, ex: infiltration, eat pawn, promotion, etc), then work backwards and label all the squares that can reach it in corresponding squares to the opponent king. If you find a triangulation point where your opponent can't step on a corresponding square  to where you are standing next move, you win. In real games you just have to remember which square corresponds to which while you calculate, mentally labelling the squares is faster than calculating individual moves.

Dvoretsky's Endgame Manual has some good talk about key squares and corresponding squares in the first chapter.