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 $m+n$.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
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 $m+n$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
92
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 $m+n$.
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