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1. Ne7+ Kh8 2. Nxg6+ Kg8 (If fxg6 3. f7+ with mate in the next move) 3. Ne2+ Kh8 4. Qg4 (or Qg3) Be3+ (the computercheck) 5. Kh1 Bh6 6. Qg7+ Bxg7 7. fxg7++
1.N-e2+ K-h8 2.Q-h6 Rg8 3.Q-g7+ Rxg7 4.fxg7#
1.N-e2+ K-h8 2.Q-h6 Rg8 3.Q-g7+ Rxg7 4.fxg7#
instead of 2. … Rg8 how about
2 … Bxh6
1 Ne7+! Kh8
2 Ng6+ Kg8
3 Ne7+ Kh8
4 Qg4 Rg8
5 Qg7+! Rg7
6 fg7 # is another line but the 7-move line was given by the first commenter.