QQ UTG 4bet pot

Poker Stars, $0.05/$0.10 No Limit Hold'em Cash, 6 Players

BTN: $17.56 (175.6 bb)
SB: $10 (100 bb)
BB: $10 (100 bb)
Hero (UTG): $11.09 (110.9 bb)
MP: $8.08 (80.8 bb)
CO: $10.52 (105.2 bb)

Preflop: Hero is UTG with Q♠ Q♣
Hero raises to $0.30, MP folds, CO raises to $0.80, 3 folds, Hero raises to $2, CO calls $1.20

Flop: ($4.15) A♥ 4♣ 8♦ (2 players)
Hero checks, CO bets $1.40, Hero folds

Villain is 3betting 12% on 69 hands sample. No occurrences regarding 4bet/5betting etc. Once unknown calls our 4bet, how do we construct our range on the Flop? Obviously our range consists mostly AKs, AQs, AKo, AA, KK, QQ, JJ in UTG when we 4bet. How do we proceed on the flop? Our villains 4bet calling range is also capped imo, like JJ+ and AKs/AKo. Do we want to cbet this flop even though that Ax is a big part of both of our range? How often is he bluffing if I check here?