While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the correct angular mode. He was lucky to get the right answer. The two least positive real values of \(x\) for which the sine of \(x\) degrees is the same as the sine of \(x\) radians are \(\frac{m\pi}{n-\pi}\) and \(\frac{p\pi}{q+\pi}\), where \(m\), \(n\), \(p\), and \(q\) are positive integers. Find \(m+n+p+q\).

(第二十届AIME2 2002 第10题)