Car not starting under 1/4 fuel level

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At first, maybe about a year and a half ago, my car would start and then immediately turn off, almost as if a security measure. As it did this, on an infrequent basis, I would notice that this would happen when the gas tank was under 1/4 tank of gas and if I put fuel in the gas tank, the problem seemed to disappear until the gas level became below 1/4 level. Then about 2-3 weeks ago it wouldn't start at all. When you turn the key, it sounds like it's trying to start, but sounds like the process isn't completing, only like half of the process is being allowed to happen (like something's not catching).