The perimeter of triangle \(APM\) is \(152\), and the angle \(PAM\) is a right angle. A circle of radius \(19\) with center \(O\) on \(\overline{AP}\) is drawn so that it is tangent to \(\overline{AM}\) and \(\overline{PM}\). Given that \(OP=m/n\) where \(m\) and \(n\) are relatively prime positive integers, find \(m+n\).

(第二十届AIME2 2002 第14题)