A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid \(C\) and a frustum-shaped solid \(F,\) in such a way that the ratio between the areas of the painted surfaces of \(C\) and \(F\) and the ratio between the volumes of \(C\) and \(F\) are both equal to \(k.\) Given that \(k=m/n,\) where \(m\) and \(n\) are relatively prime positive integers, find \(m+n.\)

(第二十二届AIME1 2004 第11题)