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While controlling an opponent, you make all choices and decisions for that player. However, because the control effect is limited to while they're searching their libraries, it's unlikely the player will be allowed to make any decisions other than what to find with the search.

You can't have the player find cards in their library that aren't asked for by the search instruction. For example, if the opponent is searching their library for a basic land card, you can't have them find a different card, such as a creature card.

If the search instruction states a quality of the card to find, such as a color or card type, you may have the opponent not find any cards. If the search is simply for a number of cards without stating a quality, you must have them find that many cards (or as many as possible, if their library doesn't contain that many cards).

While controlling an opponent, you can't make any choices or decisions for that player related to tournament rules. You can't make them concede or have them agree to an intentional draw.

The cards found in the search will be exiled rather than be put wherever the spell or ability tells the opponent to put them after finding them. Any other effects the spell or ability has will still apply. If such an effect refers to the found cards, it can't see them in exile.

If a spell or ability instructs an opponent you control to search multiple zones including the library, you decide what cards they find in all zones. They'll exile all cards they find in any of the zones.

If more than one player controls an Opposition Agent, and another player searches their library, the controller of the Opposition Agent that most recently entered the battlefield controls that player during the search. Because the last ability of each Opposition Agent is trying to exile the card and give permission to play it, the owner of the exiled card chooses which effect wins. However, that owner will be under another player's control, so that player controlling that owner actually makes the decision. In other words, the player who controls the Opposition Agent whose effect applies (and thus controls the opponent) can choose to give themselves all the play permissions unless they're feeling very generous.

You must follow the normal timing permissions and restrictions for each exiled card. If one is a land, you can't play it unless you have land plays available.

You'll still pay all costs for a spell cast this way, including additional costs. You may also pay alternative costs if any are available.

In a multiplayer game, if a player leaves the game, all cards that player owns leave as well.