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What happens when KK check-raises a blank turn?

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That's why I was planning on taking a free card off if I didn't improve. I think calling a river bet UI in this scenario is close because there are a couple of hands I beat that he plays like this (AKs, maybe KQs).

Ideally, the hand plays out like this:
turn: 3[img]/images/graemlins/spade.gif[/img] villain check calls
river: A[img]/images/graemlins/diamond.gif[/img] villain c/r and call a 3-bet [img]/images/graemlins/grin.gif[/img]

What do you think of villain's river play?