A unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and the length of the rope that is touching the tower is \(\frac{a-\sqrt{b}}c\) feet, where \(a, b,\) and \(c\) are positive integers, and \(c\) is prime. Find \(a+b+c.\)
(第二十二届AIME1 2004 第14题)