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1 Nxf5+! gxN
(if Kh5, then 2 Qd1+ and mate follows after black tosses the Queen)
2 Rxd6+ Ne6
3 RxN+ Kh5
4 Qd1 Qe2
5 QxQ#
1.Nxf5+ gxf5
2.Qf6+ h5
3.Qg5#
I mean 2.Kh5
i.e. 2…. Kh5.
1.Nxf5+ Kh5
2.Qh1+ Kg4
3.Nf3#
Minimum killing.
i.e. 3.Ne3# or 3.Qh3#
How about King g1?
1.Kg1 b2 2.Rxb2 Qxb2 3.Qxb2 Rf8 4.g4 fxg4 5.Nxg4+ Kh5 6.Nf6+ Kh6 nothing concrete is achieved.