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Hi I'm Kevin the most likely cause is the pump if the lights are flashing the the car is receiving the signal and the output only comes from pump . The fact you have also had water damage which is also very common this is the most likely cause
Ok I don't know how else I can help the pump will be the cause there is no other part that controls they are a common failure and you had water damage . Also you know the car is receiving signal .
Also if the key wasn't programmed you would have no function and we know key works as lights indicate this .if the pump had lost programming then it would also loose all function. We know pump has correct circuits as it is functioning to a degree it has to be the pump no other option