White to move. 5 seconds to solve.
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1. Rh7+
If1. . ., Kf8/Kg8
2. Qxf7 mate
If 1. . . Kxh7
2. Qxf7+ Kh8
3. Rh1 mate
If 2. . . Qg7
3. Rh1 mate
This took me a lot longer than 5 seconds, but I eventually got the solution.
Your idea is the right one, but 3.Rh1 isn’t mate in the 2. ….Kh8 line.
Also took me more than 5 seconds.