If you use Umbreon's "Shadow Drain" attack against a Pokemon with {D} weakness and a Bursting Balloon attached, what are the correct order of operations for placing and healing damage?

Umbreon's healing effect takes place within the context of the attack, so healing has to be completed before the attack is finished. Since Bursting Balloon is triggered by damage and you can't interrupt the attack's other effect (healing) to place those damage counters, you would heal Umbreon first and then place the damage counters for Bursting Balloon.

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