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Rxh6 gh (forced otherwise Rg8 mate)
Qg6+ and mate on Qh7
Rxh6+ gh (forced otherwise Rg8 mate)
Qg6+ and mate on Qh7
the third move should be better Rxh6
If the R at h1 is lazy/ill/injured and can not move 3.Qxh6+ Kg8 4.Qg6#