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Interesting hand, it's clear that villain has played his hand in a nonstandard way, whatever he has. I think a call is reasonable because the odds are good and a king just seems so unlikely. My favorite line on this river would still be c/f.
If he is any good, he simply can not call the turn here without at least AT (ruling out KQ and KJ). The only draw possible is a gutshot and nobody solid would call with that.
I would rule out KT because of flop call, this isn't a good flop to float against UTG range. AK also seems unlikely but not completely impossible.
What else is here that beats us: 44/JJ. I don't think these can be completely ruled out, although a river vshove is thin.
We also need to think what he can have that we beat. We can't assume that he called two streets on this board just so he can bluff river. He could have: QJ/AJ/AT/A4s. Would he really turn these into a bluff on the river? I don't like it because we can't offer villain bad odds and because a king is not really in our range at all.
I think he had one of these hands: AK/44/JJ/QJ/AJ/AT/A4s. On this river, if hero raised, I would shove AK/44/JJ, but fold QJ/AJ/AT/A4s. If hero checked, I would bet AK/44/JJ, but check the rest. This is why I would c/f river.
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