Let the maximum value of $\left(\sin ^{-1} x\right)^2+\left(\cos ^{-1} x\right)^2$ for $x \in\left[-\frac{\sqrt{3}}{2}, \frac{1}{\sqrt{2}}\right]$ be $\frac{m}{n} \pi^2$, where $\operatorname{gcd}(m, n)=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to $\_\_\_\_$。
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