If an Umbreon with the synchronize ability gets burned, but has a berry to cure it, does the ability proc a burn on the opponent before the status is healed?
If a Pokémon with Synchronize has a Berry that cures the status condition it was just inflicted with, Synchronize will activate before the Berry is eaten.
So yes, the Umbreon would inflict the burn before it recovers from its status condition using the berry.
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