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The quick clicking sounds like the starter solenoid keeps throwing out, then there is not enough amperage to turn the flywheel and crank the engine so the solenoid retracts the gear. As soon as it retracts it throws out again right away cause the solenoid is still got power to it, the constant throwing out of the gear by the solenoid and re-tracking due to not enough amps is what is most likely causing the clicking sound.
Most of the time this is caused by either a weak battery or sometimes a loose connection. Now, if the battery checked out good then that only leaves a bad connection. The connections include the battery terminals, starter terminals and also the ground strap and also the starter wiring itself provide a ground as well as the mating surface between the starter and engine block. So, if you have charged the battery and checked all connections and the starter bolts are tight, then the starter itself may have a short circuit.
Also, sometimes the disc inside the solenoid after arcing for sometime goes bad, you could check that and if bad, remove it and flip it over and use that side of the disc now. Also, I believe you can turn one or both of the copper bolts around 180 degrees, too.
If you want to check the starter yourself for a short just follow this rebuild procedure.