KK deep facing ace high flop and river shove

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KK deep facing ace high flop and river shove

Blinds: $0.02/$0.05 (9 Players) BN: $1.41
SB: $6.65
BB: $4.86
UTG: $5.35
UTG+1: $8.69 (Hero)
MP: $5.81
MP+1: $5.23
MP+2: $5.00
CO: $4.49
Preflop ($0.07) Hero is UTG+1 with K K
UTG calls $0.05, Hero calls $0.05, MP raises to $0.25, MP+1 folds, MP+2 calls $0.25, 4 folds, UTG folds, Hero raises to $0.95, MP calls $0.70, MP+2 folds
Flop ($2.27) A 5 4
Hero checks, MP bets $1.19, Hero calls $1.19
Turn ($4.65) A 5 4 4
Hero checks, MP checks
River ($4.65) A 5 4 4 Q
Hero checks, MP bets $3.67 and is all in, Hero folds
Final Pot MP wins $4.46
Rake is $0.19

Since villain's sizing is so strong should I have folded flop to an instant bet? I don't know why he shoves AI on the river if he wants to get value.
I think villain may have had a set of AA here, I think he could have a calling range of 99-AA and possibly 55, 44 when I limp reraise. so villain may have checked back a fullhouse here on the turn?
Can I say that the probabilities are weighted at 80% villain having AA here and 20% 44, 55 since I limp reraised? it doesn't seem we are really deep and no one is really being paid off for setmining but maybe it doesn't stop people from trying

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