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Are you shutting down completely on this hand?

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Yup.
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What do you think the liklihood is of pushing him off his Q if a K comes on the turn?

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Not that great, if he is as passive as you describe, he will check/call as always.

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How much are you normally willing to invest to see if it's possible?

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Zero, because you are drawing to 10 outs (A,K,T) and he probably isn't folding unless you push. Are you willing to put your stack behind a bluff here? I think such large bluffs are -EV in SSNL. Put your money in when you have a hand, a passive player will call your value bets then.