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1. Ne8 Qxe5+
2. Qxe5+ Kg8
3. Qg7#
1. Ne8 Rxe8
2. Qf6+ Kg8
3. Rxe8#
After considering h6 or Rd5 I’m beginning to like Ne8. Threatens Qg7# and Qf8# which leaves only black few options (Ra7 doesn’t work here). Yep, I think Ne8! wins very quickly.
1) Ne8! Qxe5+, 2) Qxe5+ Kg8, 3) Qg7#
1) Ne8! Rxe8, 2) Qf6+ Kg8, 3) Rxe8#
Ne8 Qe5
Qe5+ Kg8
Qg7#
Also
1. h6 Rg8
2.Qg7 + Rg7
3 Re8 + and mate in 4th
One more move
1.h6 loses immediately to black’s taking the rook at e5 with check since black queens the a-pawn which now covers e5 and prevents the mate:
1. h6?? Qe5
2. Qe5 a1Q
And now white must move the queen, but there is now no mate that I can find:
3. Qe7 Ra7 (Rg8 should win too)
Or
3. Qg5 Ra7 (or in this case, Bg6 should shut out the white queen)