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Solutions

(1) After 2...Qe7 3.Be4, Black wins with 3...Rxf1+ 4.Kxf1 Qe6

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This instructive move serves three purposes:
  - Keeps the Bishop pinned against mate on e1;
  - Holds the g6 pawn so Black can play ...f5;
  - Gains a tempo by attacking the White rook (so Bxg6 doesn't work).

After 5.Qe2 f5 6.Bxc6 Qxc6 7. Qc4+ Kg7, Black is a pawn up in a safe position.

(2) White's best defense to 2...Qe7 is 3.Qf2, retaining his extra piece.
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(Note: 3.Qd2 should also maintain White's advantage.
Trickier is 3.Ne4 Rxf1+ 4.Kxf1 h5 5.Qc3, which may only be good for a draw.)