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 $\_\_\_\_$
