How many ways did I butcher KT?

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Please fold the turn.

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Remember that I'm getting 11:1 immediate to call the turn with the possibility that he's raising for a free showdown.

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None of the people I would call a rock has ever raised for a free showdown. They have always had me drawing dead when some other poor soul has been in the pot and made it to showdown so that I got to see their hand when the action goes anything like this. If you want to be generous you can add AK or KQs to his range of hands, not just sets, so while you have outs, you certainly do not have enough to call.