Aq

You could even fold it on flop, that bet-size just smells too strong. It looks like some sort of protection/value bet, so you can't even know if your gut-shot hit is good.

If you had AsQx, you could consider calling flop, but on this case it should be check/fold.

what's the reason for 3betting preflop?
not saying that it is right or wrong, but it seems that you are doing it without really having much of a hand plan... which may mean that it may be better to just play it passively preflop.

as played, on that flop you either:
- just check/fold it
- cbet to try to make him fold something like 66-, AK and keep initiative. you have a gs, overcards and backdoors just in case

there's just no way that you can check/call profitably, specially vs that kind of sizing.

as for my choice i'd just go ahead and check/fold it. you are out of position, you can easily already be far behind and most likely have huge reversed implied odds with your outs.
your overcard outs can easily give villain a straight or twopair, and he just won't pay you with enough worse hands when your 4-straight or backdoor 4-flush hits.