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When Bayushi Kachiko (Atonement) plays an event from discard, at what point in initiation is she considered to have played the card for purposes of counting the 3 allowed cards? If the cost is prevented, does that still count as 1 of the 3?

Bayushi Kachiko’s 3-card limit is based on initiating the process of playing cards, so if a card’s costs were interrupted it would still count as having been played to her limit. (However, it would not be removed from the game as it did not reach step 8 of (Initiating Abilities/)Playing Cards and thus “after you play a card in this way” has not been satisfied.)