Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is \((m + n \pi)/p,\) where \(m, n,\) and \(p\) are positive integers, and \(n\) and \(p\) are relatively prime, find \(m + n + p.\)

(第二十一届AIME1 2003 第5题)