An \(m\times n\times p\) rectangular box has half the volume of an \((m + 2)\times(n + 2)\times(p + 2)\) rectangular box, where \(m, n,\) and \(p\) are integers, and \(m\le n\le p.\) What is the largest possible value of \(p\)?

(第十六届AIME1998 第14题)