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Given that villain is a known multi-tabler, and that the turn paired 3s, I think it's pretty safe to assume that villain doesn't have a set, but might have 2 pair and more likely worse. Also, given the lack of turn aggression, I think it's safe to assume that villain is looking for a cheap showdown or drawing. This essentially puts the hand range at KJ/KT or TJ. There's not many kings left, so it's actually about 50/50 as to which hand he has. Given he's a multitabler, I would value TJ more in my estimates because logically, he's probably thinking he has implied odds to call down.
What's the best river line? Well it depends on villain. If you think that if you check, villain will bet a worse hand often enough that check/call is a profitable line, then that's probably right. In this spot, I would think that bet-fold is also a good line. If you're thinking about value betting the river, the right amount to bet is the amount you would be willing to call up to if you were to go the check-call route.
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