AA on paired board

[r8ed]

I ran into this twice over the past two days. I raise preflop to 4bb and get one caller. The flop comes out with a pair and another non-scary card. I cbet my common amount and face a reraise. No reads.

Get all my money in?
Reraise? What if they push over?
Call? What if they bet/push the turn on another non-scary card?

Does the answer change if stacks are 2x max buyin?

[Pingviini]

need.more.info. What is the other player like? loose, tight, passive, aggro? how high were the cards? monotone, two tone?

[r8ed]

Both hands were seriously within an orbit or 3 of sitting down but both tables in general seemed loose with large pots > $20 based on poker client table screen.
JJ4
228

I don't have HHs but I think both were rainbow.

[Miffed22001]

Reraise both flops.
If they want to play for stacks they either have it, can read you and know you will fold even AA or are just retarded and will get stacked within 30mins anyway.
WIth deep stacks im likely to call and raise the turn looking for a cheapish showdown if it comes to that.
Only am i concerned here if opp is a rock.