… or does the card that was canceled go in the trash (or to my opponent in the case of That’s Mine), and my attempt has used up one of my plays?

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A: No, it’s that second thing you said: the card that got canceled goes away, and that play has been squandered. On the other hand, your opponent had to give up a card from their hand as well, so it’s not as though it’s without sacrifice on their part too.