Say I have an Yveltal with "Fright Night" Ability as my Active Pokemon with Bursting Balloon attached, then I play Hex Maniac. What happens at the end of my opponent's turn, when they attack and do damage to Yveltal?

Hex Maniac works until the end of your opponent's turn. At the end of your opponent's turn, you check to see if Bursting Balloon gets discarded. So Bursting Balloon places the 6 damage counters, Hex Maniac's effect wears off, and then Fright Night is reactivated preventing Bursting Balloon from being discarded.

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