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The fianchetto: keep h2 for your bishop
Against 2…g6, your plan works up to castling: 3.Bf4 Bg7 4.e3, 29 games, 53%. It goes wrong after that: the 5.Be2 d6 6.O-O Nc6 position drops to 14% over 7 games. And 2.Bf4 g6, via the other move order, is 6 games at 17%.