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1. Tfe7 Kd8 (1… Kf8 2. Ne6 Kg8 3. Re7 Bg7 4. Re7#)
2. Nc6 Kc8 3. Na7 Kd8 4. Rbd7#
Which R?
1. Rbe7+ Kd8 2. Nc6+ Kc8 3. Na7+ doesn’t seem to get anywhere. If White moves without check, then Rf4+ exchanges a pair of rooks and black’s extra material should decide the game.
1. Rfe7+ Kf8 2. N:e6+ Kg8 3. Rb8+ B:b8 4. Rg7#
1. Rfe7+ Kd8 2. Nc6+ Kc8 3. Na7+ Kd8 4. Rbd7#
1.Rf-e7check Kf8 (On 1…Kd8 2.Nc6check Kc8 3.Na7check Kd8 4.Rbd7#) 2.Nxe6check Kg8 3.Rg7 and mate next move.
There is mate in 4.
1.Rfe7+ Kd8 2.Nc6+ Kc8 3.Na7+ Kd8 4.Rbd7#
Next best may be 1.Nxe6
1. Rfe7+
A)
1. … Kf7
2. Rec7+ Nd6 (only against Rc8#)
3. Bxd6+ Bxd6/Ke8/Kg8
4. Rc8#
B)
1. … Kd8
2. Nc6+ Kc8
3. Na7+ Kd8
4. Rbd7#
I had overlooked 1… Kf8 option.
But I find in that case
1.Rfe7+ Kf8 2.Nxe6+ Kg8 3.Re8#