Rules FAQ
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In normal circumstances, the first time you win a challenge as the attacking player your claim value would be 1, the second time it would be 2, and the third time it would be 3. However, cards like Catelyn Stark (Core), Winterfell (WotN) or Rationing can prevent you from triggering the reaction on the plot, while cards like Forgotten Plans or Nefarious Acolyte can blank it. Blanking the plot after the reaction has already been triggered will not retroactively remove the claim raise though.
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If you reveal this plot with the Battle of the Trident agenda, it will be too late to raise your claim value during that challenge. But note that claim for that challenge will have been applied when the previous plot was still revealed, since the agenda triggers after claim.
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Since Battle of the Green Fork refers to its own title rather than "your revealed plot card", any claim raise from its reaction will expire if you reveal a new plot during the challenges phase. On the other hand, any lasting effect that has raised the claim value "on your revealed plot card" will remain active even after you reveal a new plot. (See References to Specific Positions in Game Areas.)