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Based on his line, he is not shoving here--in fact, I have him on TPTK, at the absolute best. If he does shove, you have 31% equity in the pot.
If you bet turn and villain c/r's all-in, he has AK
at worst. Hero has 25% equity against a range of {sets, KJ, AK}, with sets taking up the largest part of that range.
One question here is what hand are you representing and what hands are you targeting?
It's possible you have AK/KQ and possible he has J-broadway/QQ/KQ. A turn bet is ok, but assuming he calls a $100 bet I feel that those hands are reduced from his range. When you bet the river instead of checking, *your* range becomes fairly polarized to KJ, 66 and hands that are bluffing. But would you flat KJ and 66 on the flop? Has he seen you slowplay?
Many in your spot with KQ generally would take a free showdown, though it's possible you would turn it into a bluff. The right river card could be a scare card to follow through on, but with a total blank I can't think of many (better) hands he would lay down after calling the turn.