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)