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The most likely cause and is a common failure is the sending unit inside the fuel tank.
Your in luck!, On your car you do not have to remove the fuel tank for access, in the trunk there is a cover that gives access to the fuel tank sender.
Relieve the fuel pressure by removing the fuel pump relay under the hood and starting engine an let run out of fuel.
To get to the cover open the trunk and remove the floor covering, you want to make sure no carpet is left in the trunk because spillage of gasoline is a real threat and the car will smell like gasoline for weeks. Next remove the seven bolts holding the cover, then remove the cover. Next the fuel lines and wiring connectors are removed then the snap ring that holds the module down is removed, and then the fuel pump module is lifted out of the car, there will be gasoline inside so keep upright!.
The fuel level sensor is on the side of the fuel pump module and is a resitor card that the contacts wear and causes problems such as the one you are having.
Here are a few diagrams that support the text above. If you have more questions please reply back.
Rear trunk trim and carpeting.
Fuel sending unit Level sensor