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Definitely repop it preflop. You don't want to let speculative hands in for cheap preflop, because they will kill AK. Suited connectors, small pairs woudl be exactly the type of hand I would put vill on.
Lead the flop. Villain's raise is basically a bet, because the preflop minraiser just minbet. Because I put vill on suited connectors or a small pair, you probably should raise this to find where you're at. 550 or 600. Most of the time you'll get folds. If he repops you, you just have to let it go. Not worth trying to run AK through a probable set. Sometimes when villain rereraises you, he'll have something like KQ or KJ. I wouldn't be too worried about folding in these circumstances, because you'll probably take this person's money later on through their other loose calls/raises postflop.
On the turn, you're making a decision for basically all your chips because you aren't calling here and folding the river. 800 seems like a big bet for someone with a set, unless he's worried about a heart draw or 68, which isn't too unlikely for your minraise pf call, with the bb discount.
You need to raise or fold the turn. If Villain is a donk trying to strong arm you with a draw or KQ/KJ, he probably won't fold on the turn, but there are a lot of cards that could possibly kill your action (4th card to the straight, heart, ace if he has KQ, or a nonheart/nonstraight card if he has a draw). So by calling the turn and pushing the river, better hands like the set probably aren't folding, but worse hands that you want a call from may fold. By pushing the turn, you get calls from the better and worse hands, and you make more money if you are actually in the lead.
As this particular hand played out, I think it's really hard to put villain on AA. I really don't like his preflop play, he is letting the blinds in so cheap that most of the big pots he plays will be losing ones, as opposed to winning ones if he reraises and isolates it.
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