Tough runout with AK trying to get value - what do I do about this river?

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Tough runout with AK trying to get value - what do I do about this river?

SB: 102.14 BB (VPIP: 43.16, PFR: 22.53, 3Bet Preflop: 2.87, Hands: 495)
BB: 205.62 BB (VPIP: 17.39, PFR: 4.35, 3Bet Preflop: 11.11, Hands: 23)
UTG: 100.82 BB (VPIP: 25.00, PFR: 4.23, 3Bet Preflop: 0.00, Hands: 506)
[b]MP: 193.43 BB[/b] (VPIP: 45.92, PFR: 20.41, 3Bet Preflop: 9.43, Hands: 319)
[b]Hero (CO): 173.81 BB[/b]
BTN: 100 BB (VPIP: 27.31, PFR: 21.76, 3Bet Preflop: 6.02, Hands: 222)

SB posts SB 0.5 BB, BB posts BB 1 BB

[b]Pre Flop:[/b] i[/i] [b]Hero[/b] has K:diamond: A:club:

[i]fold[/i], [color=red][b]MP[/b] raises to 2.62 BB[/color], [color=red][b]Hero[/b] raises to 10.36 BB[/color], [i]fold[/i], [i]fold[/i], [i]fold[/i], [b]MP[/b] calls 7.74 BB

[b]Flop:[/b] i[/i] A:heart: A:diamond: T:diamond:
[b]MP[/b] checks, [color=red][b]Hero[/b] bets 11.11 BB[/color], [b]MP[/b] calls 11.11 BB

[b]Turn:[/b] i[/i] Q:diamond:
[b]MP[/b] checks, [color=red][b]Hero[/b] bets 22.22 BB[/color], [b]MP[/b] calls 22.22 BB

[b]River:[/b] i[/i] J:spade:
[b]MP[/b] checks, [b]Hero[/b] XXX?

I dont remember exact dynamic at time but villain expects action from me in terms of a wide but generally strong range. He is 40/20/2 F3B 25% with roughly a 50% fold to cbet on each street. I dont know much about what his 4b range would be.
Some notes:
One to build pots with hands like TP+FD, including turn raises
Keeps good hands in his river range that could normally bet for value (thinish) instead uses to bluff catch.
Intelligent knows when to bet at choppy looking boards when actually has better.

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