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you still have to put a bet out here. You are very likely to have the best hand.

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Why?

Do we fold out a better hand? Never.
Do we get call from a worse hand? Very rarely.
Are we scared of giving a free card? If villain has clubs and isn't a total moron, he has the best hand.

The only worse hands we're getting action out of here are low suited club connectors, JJ and KK. Given the J in our hand and pre-flop, none of these are very likely.

It's sucks that being OOP means that if villain also has AJ he probably wins this pot over three streets, but that can't be helped.

Of course, this is opponent dependent. Villain might be a complete spaz and read a check as us not having an ace rather than a totally reasonable play on this board with any holding but even then check calling the flop is still better than betting.