3bet SB vs MP - Facing river overbet

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3bet SB vs MP - Facing river overbet

Blinds: $0.50/$1.00 (6 Players) BN: $109.51
SB: $201.79 (Hero)
BB: $120.48
UTG: $191.64
MP: $172.96
CO: $108.00
Preflop ($1.50) Hero is SB with A K
UTG folds, MP raises to $2.20, 2 folds, Hero raises to $10.50, BB folds, MP calls $8.30
I size up a bit because I'm 172.96BB deep with the initial raiser.
Flop ($22.00) K 9 7
Hero bets $6.27, MP calls $6.27
Being deep stacked I want to proceed with more caution than normal OOP. As nut advantage becomes more relevant again with higher SPR. Generally I would check a lot OOP, K high boards being the exception. Go for small sizing because I don't have nut advantage but do have equity advantage.
Turn ($34.54) K 9 7 4
Hero bets $30.50, MP calls $30.50
After IP didn't raise the flop his nut advantage got decreased. The 4 doesn't change that (doesn't uncap). Which allows me to now start betting polarized. AK is definitly a solid value hand to do this with. If I would check this hand I likely would check/jam.
River ($95.54) K 9 7 4 6
Hero checks, MP bets $125.69 and is all in, Hero calls $125.69
I still have a very polarized range on this river as my main bluffs QJ, QT, T9, heart draws, didn't improve. Which would lead me to jamming myself. My checking range are mainly bluff giveups, with some traps in there. Randomized this hand to check 20% off the time.
Once I get jammed on I have to call any value hand I checked.

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