AJs, flop top pair + flush draw,.25/.50 NL, big pot

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You lose value on this hand if the turn comes and villian is ahead of you. Right now you are even money. Villian doesn't hate his hand, so push him in now and hope he calls. If he folds you take his $7. Sounds like a good deal to me.

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The problem with a huge overbet now is it's only called by a much better hand. Maybe an overpair (which you don't mind), but more likely only a set or two pair, both of which are favourites against you.

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Likelihood villain has a set is close to none. Most don't limp reraise 55 or 77, leaving JJ as the only reasonable hand left. Two pair? Do people you play with frequently make this play with J5, J7? Most villains will have an overpair here, and most will call with an overpair. Considering the dead money in the pot, a push is viable imo.