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Agree with baudib - its an easy check/fold on the flop.
In relation to your other thoughts, your right that generally the preflop aggressor will bet the flop as a continuation bet, regardless of whether they hit the flop or not. A common adjustment to these villians is to call their cbet and bet at the pot when they check the turn (called 'floating').
But floating here with 66 isnt optimal, because if we are already beaten we only have 2 outs or an 8% chance of improving our hand by the river. And even if we arent beaten yet, villian is likely to have a greater chance of improving his hand than we do.
So if you want to float, do it with hands where you have a greater chance of improving (more outs - such as overcards, open-ended straight draws, flush draws etc), and hence more equity in the hand..
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