Inverse Trigonometric Fun question bank (Maths)

If $k=\tan \left(\frac{\pi}{4}+\frac{1}{2} \cos ^{-1}\left(\frac{2}{3}\right)\right)+\tan \left(\frac{1}{2} \sin ^{-1}\left(\frac{2}{3}\right)\right)$, then the number of solutions of the equation $\sin ^{-1}(k x-1)=\sin ^{-1} x-\cos ^{-1} x$ is $\_\_\_\_$

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 $\frac{\mathrm{m}}{\mathrm{n}} \pi^{2}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to $\_\_\_\_$.