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I lead the flop or better c/r the flop if i think villain will fold AK.
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If he folds AK, then only better hands call and worse hands fold to your raise.
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On the contrary, if you get him to put more money in the pot drawing at 6 outs you've induced a mistake. Check/calling all three streets allows villain to play perfectly where you are often the one calling down with the worst hand drawing at 2 outs and giving him opportunities for free cards to draw out on you. The bigger the pot gets, the bigger a mistake it is to give free cards, especially with a hand that's vulnerable to so many overcards and potential draws. Raising the flop allows you to find out where you stand earlier, rather than later...
As played I think the river is a fold. The only hands you beat are 22, 44 and missed draws, however I really think that most of villains range includes made hands that we are losing to. Even the majority of pairs you beat preflop you are now behind.
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This was my thinking a week ago, but I started trying to think differently after
this thread.