Rules FAQ
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Every 3 gold means Tywin gains 1 power if there is at least 3 gold, 2 power if there is at least 6, and so on.
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You can trigger reactions like Extortion or Tyrion Lannister (LoCR) first to increase the amount of power Tywin gains. Alternatively, you can trigger Tywin first, then spend the gold to pay the cost of other reactions (like Put to the Sword).
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Cards like Casterly Rock (BtB) or Sweet Cersei that allow you to spend gold as if it were in your gold pool do not count for Tywin's ability. The gold actually needs to be in your gold pool.