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From your description, I would suspect the starter Relay or solenoid that is between the starter solenoid (located on the Starter) and the ignition switch.
ALL Connections need to be clean and tight for reliable operation, so a good cleaning of the connections could be in order. REMOVE the battery grounds before working on the large cable connections.
The location of this item is generally close to the starter, this limits the length the heavy cable required to supply adequate current to the starter. Often referred to as the castle relay, with a wire from the ignition switch supplying power to the coil and one wire going to ground. 2 large wires run between the battery and one terminal of the relay contacts. The other goes from the relay to the starter solenoid.
Should you find yourself in a no start setting and all else is working. One can jump across this relay to start the engine, BUT know what you are doing, using a jumper cable and when the engine cranks ! It is rather unnerving if you have never had to do this as you need to be under the engine in most cases !!!
Work safe AND get HELP if you are not sure of this procedure.