Check-Raise
We would get most value out of bluffs, and also if he has A he would probably bet it
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I think checking here would be a mistake, as he would know that you know he three bets in position. if the information you hold on them is correct. so they would likely check behind as that aces fills up alot of your range to call a bet at least, unless it was a overbet from them but that would be more to what they thought of your game and what they thought you thought of there game. but I dont think they would check behind on this hand as why would you even post it if it was possible.
so I think check raise is the play, as they will deffo bet big to buy the pot and then hopefully they will think that your at it cause we know they play fast and lose on the button.
Bet 1/2 pot. The villain is a "solid reg", if hero bets more than 1/2 pot the villain is likely to assume that hero is betting more because the hero felt that he lost value on the turn.
Why betting is the correct play?
Villain doesn't bet the turn even though the deuce doesn't change anything. It would seem that he has high cards.
1) Villain does not bet weak Aces because worse doesn't call him there.
2) If villain has an Ace with a strong kicker, he will raise your bet.
3) You don't look or feel silly if the villain just checks back on the river.
I am definitely thinking CHECK/RAISE here.
Pre-flop: You know Villain 3-bets often in position which doesn't send off any red flags.
Post flop: Your check call is best line. It is safe to assume Villain is capable of making this steal attempt without a made hand, given your knowledge of his aggressiveness.
Turn: His check does indicate that he may have a very weak hand like possibly a pair of 9's or an underpair, maybe some Ax type hand. Given his LAG image, he would likely bet any sets or overpairs given your flop call.
River: So far, Hero has played very weak and passively during the hand. Any bets here will cause a likely fold by the Villain as he doesn't seem to have a hand worth calling with. By check/raising, you allow the Villain to take a final stab at the pot and get some value from your set. If he spiked an A that would be your best outcome for maximum value, especially since its extremely unlikely he has your hand beat. AA wouldn't make sense as his holding in this spot so there's no cause for concern if he decides to put up a fight when you raise. A raise of about 2x to 2.5x his bet on the end may invoke a call from Villain with a weak A or even a pair of 9's given how you've played the hand.
Since everyone already suggested the c/raise option, I'm going to put another idea out there.
I would go ahead and try to induce with a 1bb donk bet.
Since a big part of villain's range has an Ace in it, especially after his turn check back, and we know he's an aggro reg, we can expect him to go for thin value with his weaker aces and ofc go for value with his big aces too. Another thing is that he might interpret our 1bb donk as a block bet and turn his hand into a bluff which also goes in favour of the 1bb donk bet.
Ship that upgrade this way 
Hi!
I think the best way to play this hand would be to lead and bet like 50% he would probably raise if he would have AK or two pair, and when he does raise we can shove.
*By betting we don't loose value from weaker hands.
*Since we bet half pot - 38.5$ - pot would be 117.5$ and if he raises to 140$ we would be able to shove. Pot would be 257.5$ he would have 328,72$ left , and if we would shove pot would be 604,22$ and he would have 1.8 : 1 pot odds.
I would lead out 3/4, If he's got an ace he'll call or raise , If he's empty he'll fold and if he's got a weaker pair he will check if we check to him for a cheap showdown.
I won't let him get the possibility to check behind. And if he raises , of course we shove
If i was playing, i would have tried something different. We could just min raise, so he could re-raise us or do something stupid like shove. Check-raise is good option though, since there is 90 percent chance ( in my opinion) that he has an ace 
The As is in the range of vilain.
1)check/raise: 2,5x the value bet of vilain and call if vilain shove after this
2)value bet thin for induce a raise of vilain, and if I raise, pay then.
Short Answer: Check/Raise. For sizes check the bottom of the post.
So, we are ahead. Only 54 and AA beat us and both probably would have bet the turn. Plus, only 3 combos of AA and an already unlikely 54 preflop (I'd give it only 4 combos of the suited ones).
We therefore exclude check/call and check/fold as we would under-represent our nuts, as well as bet/fold and bet/call (a slight doubt on the latter in case of fancy plays with AA/54, since we are deep, but not really).
- Bet/Raise
Villain can have random bluffs, Ax, mid-pairs.
Normally, I assume, we wouldn't bet here that much since villain is full of Ax and there are not many missed draws. So our perceived range should be either nuts or pairs turned into bluff.
Only bluffs would raise you, and hardly ever, since villain is a solid reg.
Weak pairs will sometimes find a hard call but most will fold, Ax will just call and never turn into a bluff. Probably even A9/A8 wouldn't.
The thing is, probably Ax would bet themselves as well. So there's no added value from them, since we can't raise. - Check/Raise
After showing weakness 3 streets in a row we offer villain the chance to thin value bet with weak pairs, or to turn them into a bluff, and obviously we also get all the random air bluff bets (and if xe likes to 3b a lot IP, xe might have a lot of them). Ax would bet, and now we can raise.
Overall, Check/Raise has more value.
For the size, it depends, if xe bets under 1/2 pot I'd go for 3x at least, or 2.5x for anything between 1/2 and potsize, and shove the rest. These sizes are to possibly make xem think xe can still produce Fold Equity by 3betting us.
I will Donk 198bb after 4 seconds of thinking ! the times that he will call with good Ax will cover all the miss value....If I x/r maybe he will fold AQ/AJ and even AK (if there is no reason to think that I am a monkey bluffer).
cheers !
Check-raise
To extract some more money from bluffs and not only from Ax if we donk bet.
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The river perfectly hits his range. So there are two options:
1) Lead out.
2) Check-raise.
His range doesn't have many overpairs on the flop, since I would expect a solid reg. to fire again on the turn. This means Ax with x>T is the biggest part of his range, with a smaller one to have some KQs and KJs hands. I would most definately expect a solid reg. to bet this river when checked to. If he is willing to bet, I think it wouldn't expect a bet more than 2/3 of the pot and I think he will get away most of the time if I check-raise, unless my image is not a nit's one. So I would prefer leading out around 90%-110% of the pot (70$-80$) to induce a call from any Ax he may be holding. Sometimes he may have caught aces-up as well (that checked the turn with his middle-bottom pair for pot control, i.e. A8s). The sizing may also seem tricky and will make my range polarized enough to receive a call quite often.
All in all, I'd say the hero bet $70-$80.
