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When Eager Construct enters the battlefield, each player may scry 1. (To scry 1, look at the top card of your library, then you may put that card on the bottom of your library.)
If multiple players are instructed to scry at once, such as while resolving Eager Construct's triggered ability, those players each look at the top card of their library at the same time, then they choose in turn order whether to put those cards on the top or bottom of their library.
