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I can't see too much that you could have done to alter the outcome of this hand, in all honesty.
You are right that you really don't want to push pocket 3's too hard pre-flop, and with his A5s, he would have called any modest raise, any ways. Your opponent flopped the wheel straight, and had to think that it was the best hand, assuming no one was holding 5-6. I doubt you are going to get him to lay that hand down, even with an All-In bet on the flop. At best, he probably would have pinned you on a small set or two pair, which he already has beaten. He slow played his straight on the flop, and then pushed in a near pot-size bet with the 7 showing on the turn. He was trying to trap a lesser made hand and he got you with your set.
That's a tough read, and maybe the only hindsight was that you you could've made a smaller (non-A.I.) re-raise to see if you'd get a call, but if you thought your trip 3's were good, then the 9 on the river wouldn't have changed anything, and you probably would have bet your remaining stack on the river figuring you had your opponent trapped with two pair or something.
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