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Meh. I don't think it matters much given the stack sizes.
I feel like his range is heavily weighted towards a T given the flop action and his stats, but you have to condition that by the fact that there are twice as many ways for him to hold Kx as Tx. If you feel that you are even money to be tied, you can bet enough to put him AI and make a profit if he can ever fold a T getting 3:1 on that board (probably not).
I think the major decision is whether he raises a hand like Kx on that flop and then calls a reraise that reps AA, AK, KK or Tx.
I think check/calling is terrible. I would always push if I were going to call, and I would only check if I thought he would only bet the turn with a K, in which case I would check/fold. If he checks behind on the turn, I would make a bet that was just enough to put him all in (as opposed to pushing) on the river.
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