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Villain: 123 hands, 30/15, AFq: 51%, Flop raise: 18% (3/17) Fold to flop cbet: 63% (5/8), WtSD: 19%, W$SD: 50% (3/6)

Ongame Network $0.15/$0.30 No Limit Hold'em - 6 players - View hand 1778909
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UTG: $27.27
MP: $40.85
CO: $37.56
BTN: $37.53
SB: $29.70
Hero (BB): $35.34

Pre Flop: ($0.45) Hero is BB with A A
1 fold, MP calls $0.30, CO raises to $0.60, 2 folds, Hero raises to $2.10, 1 fold, CO calls $1.50

Flop: ($4.65) 2 K K (2 players)
Hero checks, CO bets $3, Hero calls $3

Turn: ($10.65) 7 (2 players)
Hero checks, CO checks

River: ($10.65) J (2 players)
Hero checks, CO bets $32.46 all in

Hero?

I think how I played and given that there is only 10.65$ in the pot without his all-in, it is a fold. My question is: What is the best way to play hands on duplicated board with an overpair? I hate these way ahead way behind situations. Plus the flop is twotone, so the villain can raise not only kings, but FD-s too, or can bluff of course, or there is a chance that he floats or simply calls with his kings, and I have a hard time on the turn. Against passive players this isn't a problem, but agressive players can cause pain. I know, that he have less chance to hold a king because there is two kings on the flop, but I think it's still not an easy situation.

I probably 3bet bigger pre, like $2.5 because we are oop and there's still a limper.
OTF I just cbet and expect to get called by pairs, KX, FDs and maybe even A high (but unlikely since we have AA and its very villain dependent)
OTT I'd just value bet again