Given that $$O$$ is a regular octahedron, that $$C$$ is the cube whose vertices are the centers of the faces of $$O,$$ and that the ratio of the volume of $$O$$ to that of $$C$$ is $$\frac mn,$$ where $$m$$ and $$n$$ are relatively prime integers, find $$m+n.$$

(第二十三届AIME2 2005 第10题)