Nut flush draw gets c/r on the turn

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Nut flush draw gets c/r on the turn

Blinds: $5.00/$10.00 (6 Players) BB: $1131.75
UTG: $1866.66
MP: $1150.25 (Hero)
CO: $1000.00
BN: $2830.80
SB: $824.00
Preflop ($15.00) Hero is MP with A J
UTG folds, Hero raises to $30.00, 2 folds, BB calls $20.00
Flop ($65.00) K 5 6
BB checks, Hero bets $45.00, BB calls $45.00
Turn ($155.00) K 5 6 4
BB checks, Hero bets $108.50, BB raises to $352.00, Hero calls $243.50
River ($859.00) K 5 6 4 K
BB checks, Hero checks

Hi,
I'd like to get some peoples thoughts on this hand. Villain is pretty much unknown. I only know that he is a 5/10 reg so i'm guessing he's pretty good, hand reading blah blah blah.

My plan here was to barrel the river on pretty much any card apart from a K probably. When he c/r I think he's pretty much always weighted towards value (so flushes, the occasional 78s). Having the Ace of hearts myself his bluffs are pretty limited so i guess AQ with the queen of hearts is the only one I can think of unless he's turning things into a bluff which seems pretty unlikely in these positions. We were 110 bbs effective at the start of the hand.

So my questions are:
1. Can we even call the turn getting 28% here? Our implied odds must be fairly limited I would think. I assume we pretty much always have 7 outs. I would be flatting my whole range here so i guess that would be all sets, all flushes, 78s and then i'm unsure about this part of my range.

  1. What is our plan for the river? If he checks on a heart which bet size do we use? If he checks on a blank do we bluff? We have so few air hands for getting to the river this way that I would think we need to bluff here. I just can't believe he's going to fold any of his range here apart from the AQ i mentioned earlier.

I was pretty lost on the turn and river so any input appreciated.

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