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Right, but ...

  1. Ne3 Re1 ... what to move now?

Or also

  1. Ne3 Qe1 what now? :)

Let's see:

  1. Ne3 Re1
  2. Rh2 a1=Q
  3. d8=N Qa7
  4. Ng4+ hxg4+
  5. Kxg4#

and

  1. Ne3 Qe1
  2. Ng4+ hxg4+
  3. Kxg4 (White king is very safe on g4, black can't stop d8=Q) a1=Q
  4. d8=Q Black can't do anything!

or
2.. Ng4+ Kg5
3.. Nh2 a1=Q
4.. d8=N! Black cant' stop checkmate on f7 and e6 at the same time

I mean after

  1. Ne3 Qe1
  2. Ng4+? hxg4+
  3. Kxg4 Qxh1 black wins.

Lol sorry. Now I see the problem xD

  1. Ne3 Qe1
  2. Rxe1 Rxe1
  3. Ng4+ hxg4+
  4. Kxg4 a1=Q
  5. d8=Q Qa8
  6. Qxa8 White wins, black can't stop checkmate on h8
  1. Ne3 Qe1
  2. Rxe1 Rxe1
  3. Ng4+ Kg5!

Just make it complete now. :)

  1. Ne3 Qe1
  2. Rxe1 Rxe1
  3. Ng4+ Kg5!
  4. Nh2 a1=Q
  5. d8=N Qa7
  6. Nf7+ Qxf7
  7. gxf7 g6
  8. f8=Q gxf5
  9. Qg7#

Are you sure you didn't forget any moves?

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