Can Cypher copy ATA's? If so what happens if the other player has Thunderbolts?

Also when using an ATA like Thunderbolts at what time do you declare what team ability you will use?

If it's a timing thing and Cypher could copy ATA'd powers (like mystics from Thunderbolts) and both players were playing the Thunderbolt's ATA and a Cypher, that wouldn't mean that one player goes first and declares their ATA (say mystics) but the other players hasn't chose their ATA yet, wouldn't that leave one of the players at a big disadvantage?

I hope that this makes sense. It's late here. Thank you in advance.

Yes, you can copy ATAs.

And here's the entry for Cypher from the PG:

Ok thank you. But is there a timing issue here? Would both players have to select which Thunderbolts TA they were going to use first and then say which TA Cypher was going to copy after? Or would a person do it when they place their figures? I'm just tryign to understand the interaction here.

The first player chooses, and then the second player chooses.