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I think one possibility is playing this like a straight up gutshot draw. The flop is a free card raise. Take the free card on the turn maybe?? I don't know.

If you think you are behind, you got enough to at least call to draw to a gutshot. You could raise to get rid of any other kings, and maybe take a free card. I think this option is at least better than folding.

Now are we ahead enough to bet the turn? I'd say no, but I'm no expert.

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I agree that the hero's chances of being ahead are quite dismal on the turn, but in this large pot he only needs to be ahead here a small percentage of the time to make betting correct, and plus you never know, maybe someone will make a horrible fold with their ace bad kicker. I know this possibility is remote but no one can make a bad fold if the hero doesnt bet. So between the small possibility the hero is in the lead and the small possibility the hero can force a better hand to fold, I think betting the turn is better than checking, but I admit this is a razor thin decision and I would never think checking is a bad play here by any means.