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Hello, so far it sounds like the compressor contactor is 13 years old and has burnt contacts. ($70 item) or needs a hard start kit installed or both, ($100 item), however I am not clear yet on *what is taking 4 or 5 attempts to start... the compressor, or the fan motor.
If its the fan motor with the problem, it could have worn out bearings by now, or a weak run capacitor.
We can go from there.
You will need a clamp around type amp meter to diagnose this effectively,
$25 at harbor Freight, $50 at a big box electrical dept. Check the amp draw on each wire separately going to the motor as it is attempting to start... and the run capacitor. That will isolate the problem either to the relay on the printed circuit board or the motor itself.