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If the icemaker won't cycle on it's own but it goes through 1 cycle when you do the manual test after pushing in the paddle 3 times, either the freezer is not below 10 degrees so the icemaker is refusing to cycle, or your icemaker has just failed and will need to be replaced. There are parts sold separately for the icemaker so it doesn't matter exactly what is wrong internally with the icemaker. GE decided to go with a more universal icemaker for their refrigerators to keep the cost of the icemaker down---the downside is that the icemaker itself is not serviceable.
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If the motor on the icemaker is frozen, then the icemaker is leaking and while heating it up might temporarily fix the issue, the motor will freeze again and you'll be in the same problem in a couple of cycles. As for charges---I have absolutely no access or idea what you are signed up for or how you are charged. I'm am just an appliance expert with factory training. You would need to speak to customer service about any charges and whatnot.