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Per the FAQ:
Q: Must the Basic Character effects be applied *immediately* following the successful completion of a Mission or may they be used at any time prior to the end of your turn?
AB: The effect must be used immediately, with the exception that if multiple effects happen at the same time, you can choose in which order to apply them. Such as multiple Basic Characters in your bridge plus a Reward from a Mission you just completed.
Does this mean that with a Mission reward that includes +X Search, those Searches must be performed before continuing the turn?