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)