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I was initially going to say that you should make a smaller raise preflop. My logic for that was that you would have easily denied villain any implied odds to call with a >$50 raise.
I thought about it some more and like your raise because you are really repping K-K/A-A now, sometimes A-K.
Villain called. If he isn't a thinking player (or believes you are bluffing) he could have Q-Q+, A-K+. That seems like a reasonable range for him to call you 4-bet, plus a series of weaker hands he may have called with to see how you would play the flop. So, against his range we are in trouble, and it is highly likely only a bluff will get us to win this pot.
Flop comes with an A and 2 blanks. There is a lot of money in the pot (~$270). If he is bluffing he may bluff raise over a probe sized bet if you chose to do that, since he would know you will fold any hand that doesn't have an A in it, also probably a 50/50 considering your range of hands. He may even call with K-K until he is pot-committed, hoping to catch you bluffing. He will never fold A-K or A-A, so this bet will do nothing but potentially build a pot in which we are almost certainly WB, or going to get muscled out of.
I think you have to make a regular bet and hope to not get called or raised. A check wouldn't get you any information and you would lose control of the betting Even though there is only 1 meaningful bet left in the hand, you have to be the one making that bet. A 1/2 pot bet makes a bluff raise by villain less likely(since he would know you would be getting over 5 to 1 on your call if he bluff raised). He would also have to fold K-K, Q-Q to this bet, unless he was resolved to his assumption that you were bluffing (also unlikely since you 4 bet preflop and now led out with a bet on an A high flop).
A shove is worse than a reasonably sized bet, since you don't get an opportunity to fold/raise if villain is indeed holding an A, and both bets accomplish the same goal (finding out if villain has an A).
With so much money in the pot and villains 50/50 range of hands he would fold/ shove with, i think it all makes a bet automatic.
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