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What happens when I play Procession and use Throne Room as an action card affected by Procession?
Let me give you an example. I have a hand of:
I play Throne room within Procession and then:
Do I play only one action card in Throne room, e.g. Bridge, and it is played 4 times (Throne room played twice, both with Bridge inside) or
Do I play two action cards in Throne room, e.g. I use Bridge in first play of Throne room and then Bridge is used and when I play Throne room for the second time I have to use Village (my only action card left)?
In other words: does Procession treat an action card in Throne room as one card (as we usually do when we play Throne room) or not?
If you Procession a Procession, you will play one Action twice, trash
it, gain an Action card costing more, then play another Action twice,
trash it, gain an Action card costing more, then trash the Procession
and gain an Action costing more than it.
There is no reason to assume that Throne Rooming a Procession would behave any differently: they both allow you to "play an Action card from your hand twice".
So your second scenario is correct: you would play two different action cards twice, not one action card four times over.
Regardless of whether you throne room your procession, or procession your throne room, you'll play the bridge and village twice if you play them. Some key things to note:
If you Throne Room first you will play Procession twice, trashing both the village (getting a 4 cost) and bridge ( getting a printed 5 cost card, see below.)
If you Procession the throne room, you will play the village and bridge twice, and then trash the throne room (getting a printed 5 cost card, see below.)
Bridge's effect will go off twice, reducing the cost of all cards by two, but this is largely irrelevant, unless you play a BUNCH of bridges, or procession something really cheap like Poor House (cost 1). If you do manage to play enough bridges, to reduce the processioned card to cost 0 you'll then trash it for something costing 1, which can get wacky if you play enough bridges.
You do not have to reveal your hand but only procession includes a "you may" clause. If you Throne room your procession you don't have to play the bridge or village if you don't want to. You could throne room your procession, playing procession twice for no effect if you wanted. However if you Procession your throne room, you would have to play 2 action cards twice for throne rooms effect.