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There has to be a checkmate somewhere.
1. … Rg6+ 2.Kh1 Qxf2
1. … Rg6+ 2.Kf1 Be2+ Qxe2 3.Qh3+ Ke1 4. Rxg1+ Kd2 5. Rxe2+ and 6. Rxc1
1. … Rg6+ 2.Qg3 Qh3+ 3.Kh1 Bf3+
1. … Rg6+ 2.Kh1 Qxf2
… continuing:
3.Rxg6 Bf3+ 4.Qxf3 Qxf3+ 5.Rg2 Qxa3 6.Rcg1 Re7
Black’s definitely better, but does he have a forced win from here?
1. … Rg6+ 2.Kf1 Be2+ Qxe2 3.Qh3+ Ke1 4. Rxg1+ Kd2 5. Rxe2+ and 6. Rxc1
Something garbled in those move numbers. Did you mean:
2…Be2+ 3.Qxe2 Qh3+ 4.Ke1
…? It looks like:
4.Rg2 Qxg2+ 5.Ke1 Rxe2+ 6.Bxe2 Qg1+ 7.Kd2 Qg5+ 8.Kd3 Qd8
… puts up more of a fight than 4.Ke1, although Black still has a big edge — enough to win?
1. … Rg6+ 2.Qg3 Qh3+
Black actually has a mate in 5 with:
2…Rxg3+ 3.fxg3 Qe4+ 4.Kh3 g5 5.g4 Qf3+ 6.Rg3 fxg4#.
How about
1. … Rg6+
2.Kh1 Be2 forcing white queen to move followed by
3…Qe4+
1…Rg6+ 2.Kh1 Be2? 3.Qxe2!