Thin rive bet?

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Posted by posted in Mid Stakes

Thin rive bet?

Blinds: $2.50/$5.00 (6 Players) BN: $750.01
SB: $708.05
BB: $767.55 (Hero)
UTG: $427.50
MP: $638.74
CO: $513.83
Preflop ($7.50) Hero is BB with 3 A A A
4 folds, SB calls $2.50, Hero raises to $15.00, SB calls $10.00
Flop ($30.00) 5 4 K
SB checks, Hero bets $20.00, SB raises to $55.69, Hero calls $35.69
Turn ($141.38) 5 4 K K
SB bets $72.06, Hero calls $72.06
River ($285.50) 5 4 K K J
SB checks, Hero checks
Final Pot SB wins and shows three of a kind, Kings.
SB wins $282.70
Rake is $2.80

I think given the assumptions that betting the river in villains shoes (K876) would be too thin, but if he instead had 5576 a river bet might be better than check folding (whether it is better than check calling is another question)?
Reads: good reg

Image 1: Villains range
Image 2: My range vs his hand
Image 3: My range when he has 5567

Any thoughts?

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