Fuel filter and/or perhaps the fuel pump relay. You could jump the fuel pump from the battery to test the relay, but its not a good idea to run it that way. The relay is there so if the motor dies the pump stops.
(As in: you get into an accident, the fuel pump won't keep pumping gas on an engine compartment fire.) It really does sound like a fuel problem to me as well.
The other possibility that hasn't been mentioned is the coil resistor, but since it does kind of continue to pop after you quit cranking it over. In my experience if that resistor is toast it will run fine while you crank the starter but die dead cold when you let off the key to "ON". So, I doubt it is that, but if you run out of ideas that is another place to look.
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