Ax flops in 3bet pots as 3 better are not good spots to plan a c-raise unless vs very specific opponents.
People expect you to cbet and check back more often then not.Then on the turn you never know where you stand if you are raised...exactly what happened here.
I dunno what to do,depends on villian stats.

Don't like the x/r plan. what other hands in our range would we 3bet and x/r here ? i think that bet/bet/bet is much better. the way you played dunno i would probably call and x/f river.

Why on earth would you jam flop? That's going to be a pretty big overbet not?

Also I call here, and check call most rivers just because checking the flop here then leading turn makes me feel like you have no value hands in your range. He might think that too.

How did the river play out?

Why check raise? if you dont know the answer then its terribly bad

3barrel? i'm assuming they 4bet AK a good% of the time from those positions. also think flatting pre might be better but squeezing is fine when there is a fish.

c/r is really bad imo, villian will at least be checking back all pp's here, and not really betting much of their air when checked to, and sometimes they wont even bet their AJs, you look like you have marginal-decent hand strength when you check the flop here.

Your C/r can basically only get value from AJs/ATs and flush draws. And i think opponents will check their AJ/AT back on flop a decent % of the time.

the reason you wanted to check jam is because when you say it out loud it sounds sexy

Hate the flop play, now click it back and shove river