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1.Bg2 Kc7 2.Ke7 c4/Kc8 3.Qb7#
1.Bg2 Kd7 2.Qb6 c4/Kc8 3.Bh3#
1.Bg2 Ke5 2.Qe3+ Kf5 3.Bh3# or 2…Kd6 3.Qe7#
1.Bg2 c4 2.Qb6+ Ke5 3.Qf6# or 2….Kd7 3.Bh3#
Nice little puzzle. Yes, 1.Bg2 allows access to h3 in some lines wihout taking the f-file away from the queen in others.
Unfortunately, I had only eyes for 1.Kf6 taking the e5 square away. That is a mate in 4, I think.