If I use Absol's "Cursed Eyes" Ability to move three damage counters from one of my opponent's Pokemon to another, what happens if I choose a Pokemon with the "Theta Stop" Ancient Trait?

Since Theta Stop prevents all effects of Abilities done by your Pokemon, if you choose to move the damage counters to a Pokemon with Theta Stop the damage counters would be discarded rather than moved onto that Pokemon. However, if you choose a Pokemon with Theta Stop as the source of where the damage counters would be moved from, they simply are not moved and the Cursed Eyes ability stops there.

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