An angle \(x\) is chosen at random from the interval \(0^\circ < x < 90^\circ.\) Let \(p\) be the probability that the numbers \(\sin^2 x, \cos^2 x,\) and \(\sin x \cos x\) are not the lengths of the sides of a triangle. Given that \(p = d/n,\) where \(d\) is the number of degrees in \(\arctan m\) and \(m\) and \(n\) are positive integers with \(m + n < 1000,\) find \(m + n.\)

(第二十一届AIME1 2003 第11题)