Why is 11.Bxe4 bad and what is White's best move?

Thought process:
Checks - Mate Threats - Captures - Threats
The attack is simple. The defense requires more thought, but if you examine checks first, the solution pops right out at you.
11.Bxe4?? Bh4+ 12.Bf2 Qxf2#
11.Qh5+! g6 12.Bxe4 Qxh1+ (...Bh4+ loses) 13.Bxh1 gxh5.
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