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First - yes you need to re-raise preflop. Even with you're position, AJ just doesn't play well four-way. Hands like this make one-pair hands post-flop, and those play best against fewer opponents. This is especially true in this situation because you know there is a good chance your hand is best. If there were tighter players in the hand, you could just call.
You want to be able to use your position to win the hand if both you and your opponent miss the flop. Inducing folds pre-flop, and making a statement about your hand strength pre-flop accomplishes this.
Also, the benefit of being on the button is information. Re-raising pre-flop gives you a cheap look at valuable information about all of your opponents hands. I would argue that if you had raised to $15 pre-flop, you would know alot more about your situation after the flop, and still have put in the same amount of money.
As played, I move in on the turn. He's probably not folding any Ax hands given his remaining stack. As mentioned already, AK, AQ, and A8 seem unlikely given the preflop action.
I could be wrong, this is just my playing style. But in this spot, I wouldn't want the pressure of thinking about things, I would rather the pressure be on him. I would just raise him, let my opponent be the one who has to make the decision for all of his chips.
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