I had the same problem a couple of years ago. I replaced the solenoid. It worked for awhile then it started clicking like you said. The solenoid has a plunger in it & it can go bad & act like that. I took the solenoid back & got another. It still works fine.
I just replaced the solenoid in my F150 too. If you turn the key & get no sound, it's usually the solenoid.
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