KK on A high +BDFD

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KK on A high +BDFD

Blinds: $0.01/$0.02 (6 Players) BN: $2.00
SB: $3.45 (Hero)
BB: $2.39
UTG: $2.15
MP: $4.52
CO: $2.00
Preflop ($0.03) Hero is SB with K K
UTG raises to $0.06, 3 folds, Hero raises to $0.24, BB folds, UTG calls $0.18
Flop ($0.50) A T 6
Hero bets $0.32, UTG calls $0.32
I think board favours both, but I bet with the intention of c/folding turn
Turn ($1.14) A T 6 J
Hero bets $2.82, UTG folds
Not sure if this line makes sense after betting flop. I was about to fold but now I have 12 outs to the nuts and I dont really want to c/c only to give up river.
If I give villain a range of AQo+,AJs+, I still have 26% which is borderline, but with shoving I have fold equity too, right? am I ranging villain too optimistic/pessimistic? thanks
Final Pot SB wins $1.10
Rake is $0.04

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