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I think there should be a limit on the hand size you can underdeck to get the set aside cards (maybe 3?), otherwise it's just a drawing card that grants you a way out of RBD and extra cards for a potentially low cost.
You could make it like a set aside card though where u may under deck one for one at any time.So under deck 3 take 3 from set aside or 7for 7 or 1for1 ect.
Version 2:Artifact (no name or verse yet)"Discard to reveal 7. Put lost souls in play and set the rest aside. At any time during a future turn, you may underdeck a card from hand to add one of those cards to hand."
This sounds like you can only do one card at a time. Had you wanted to make is so that you could exchange 1-for-1 up to 7?