Bloodthirsty Adversary

Haste
When Bloodthirsty Adversary enters, you may pay {2}{R} any number of times. When you pay this cost one or more times, put that many +1/+1 counters on Bloodthirsty Adversary, then exile up to that many target instant and/or sorcery cards with mana value 3 or less from your graveyard and copy them. You may cast any number of the copies without paying their mana costs.

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Haste
When Bloodthirsty Adversary enters the battlefield, you may pay {2}{R} any number of times. When you pay this cost one or more times, put that many +1/+1 counters on Bloodthirsty Adversary, then exile up to that many target instant and/or sorcery cards with mana value 3 or less from your graveyard and copy them. You may cast any number of the copies without paying their mana costs.

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After you pay the {2}{R} cost one or more times, a second ability triggers and you choose the targets for it. Players may respond to the new ability at that point. As that ability resolves, Bloodthirsty Adversary gets its counters and the target cards are exiled.

Copies cast this way are cast during the resolution of the ability. You don't have to follow normal timing requirements and you can't wait and cast them later.

Whether or not you cast the copies this way, the original cards will remain in exile.