Turn and River play, AA OTB

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How can you put this guy on a flopped set with a board of KJ676?

Also, how can you put him on K6 specifically? How is that more likely that KJ or K7?

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I think a set and KJ are likely in that they would be scared by the flush coming in when they call the 3-bet. The only reason I suggested K6 is that Dude donks us on the river. Dude's whole line is very confusing at best. I can't see him limping JJ, but there's only one 66. I think there could be 2 reasons for the S&G: he's bluffing or he has a monster. I admit that KJ is likely, which leans me towards this river call.

However, doesn't SSH suggest the following line if it goes A->Hero->B, and we believe we can beat A (or A is on a bluff), but we can't beat B: we should raise the river to force B out? I'm not sure it applies however, because if B has a flush, he's not folding.