With AKs I would sometimes 4 bet and sometimes call a 3 bet, or even sometimes call an open raise with it just to mix up the play a little. I've learnt not to over value AK as it has gotten me into trouble so many times before. Preflop you played absolutely fine here. I'd be a little wary of the 3bet though as they can usually represent hands like AA and KK.
Post flop I would have led out with a cbet instead of checking because of the flush draw that is there and it would also have given villain an opportunity to fold or to raise over the top of you which would let you know that you were probably beaten and you could have found it easier to throw the hand away not having invested so much of your stack in the pot. River for me is a fold here as well but a hard one tbh. I'd be annoyed folding it but would have to convince myself I was definately beat.
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