4-bet AK when out of position. Just makes your life easier in so many situations. Plus, his 3-bet size was pretty small, as others have said.
As played, it was very hard to put him on a hand. I would never have guessed pocket 4s. Going off standard 3-betting range, he doesn't have too many kings in his hands here, AK is 3-betting, maybe KQ if he is aggressive and 3-bets a lot.
Checking the flop will widen his range and allow him to c-bet bluff you. I think you make more money in the long run that way.
Turn I think you should bet out for value. You don't really gain much by checking, and you don't want to give villain a free card.
River I would call. As villain played it, it looks like a worse two pair, maybe a set of Tens that floated you flop. You also lose to QJ, but I don't that's too common here. That might bet out on the turn.
4-betting pre would narrow his range and make playing him a lot easier.
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