A container in the shape of a right circular cone is 12 inches tall and its base has a 5-inch radius. The liquid that is sealed inside is 9 inches deep when the cone is held with its point down and its base horizontal. When the liquid is held with its point up and its base horizontal, the height of the liquid is \(m - n\sqrt [3]{p},\) where \(m,\) \(n,\) and \(p\) are positive integers and \(p\) is not divisible by the cube of any prime number. Find \(m + n + p\).
(第十八届AIME1 2000 第8题)