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If only the rook were on g3 in the position, then Ng6+ followed by Bg5#, so we take it there!
1. Rg5+ Kh6 [ … Kh4 2. Bg3#]
2. Rg3+ Kh5
3. Ng6+ Kh4
4. Bg5#
read Ng7+ for Ng6+, of course.
1. Rg5+ Kh6 (otherwise mate in 2)
2. Rg3+ Kh5
3. Ng7+ Kh4
4. Nf5+ Kh5
5. Rg5 mate