Queen Endings are like a Maze

Queen endings are like a maze to me.

So many possibilities, checks flying everywhere ... diagonals, files, ranks. Where do you start? How to make sense of it?

From my endgame training, I'm trying to switch my thinking in terms of schemes rather than moves.

Averbakh 1962

When I first encountered this problem, I didn't know where to start ... the King has 6 possible moves.

This was the wrong way to go about handling this problem (brute force is not the answer).

One must think schematic ... what situation do you want to set up.

Hiding the King pawn

It's worth exploring bringing the King to say h7 or h8. However, this rarely works well as as the Black Queen checks from all angles with the White King getting in the way of the pawn

The Cross Check

This idea is to exploit the position of the Black King so that when the Black Queen checks, the White Queen interposes with a check. 

To this end, White actually wants to get his King to f2 and interpose any check at h2 with Qg2. This way, when the Black Queen nexts checks, the White King will move and deliver a discovered check along the 2nd rank from his Queen.

With this in mind, see if you can solve

Kling and Horwitz Chess Studies, 1851

I first came up with the idea of bring all the way to e1 so that I could interpose any check with either Qd2/Qe2/Qf2 and a cross check

This exercise alertered me that there are two ways to skin a cat ... another idea is to move the King a-file. That way any check along the a-file can be interposed with Qa7

Can you solve?