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Intermittent problems are very hard to diagnosis. I believe it is a fuel pressure problem and possibly the fuel pump not keeping the fuel rail pressurized for start-up. Unfortunally the only way to properly diagnosis is to have a fuel gauge on the vehicle when the car doesn't start or chokes out.
I was going to suggest replacing the fuel pump. But you do run the risk of it not fixing the problem. But I am confident the pump is the cause. Replaing the pump will run around $500. You can try replacing the fuel filter first as it is cheap to do so, but I do not beleive it is the problem. You cannot tell if the fuel pump is bad by just looking at it.