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That sounds like the car has a bad crankshaft position sensor. This will cause the car to lose position, the motor looses spark and the car turns off. You can confirm this by doing a quick spark test when the car turns off and refuses to restart. If there is not spark, the sensor has gone bad.
Crankshaft position sensor : SU4101 $34.99
I would not suspect the pump. If when the car is running you are having no problems, I would not suspect the pump.
Do a spark test when the car shuts off and does not want to start. I am sure what you will find is that the car does not have spark, but has fuel.