Mathematics Question from JEE Main 2026 · 28 January, Shift 1

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 $\sin^{-1}(kx-1)=\sin^{-1}x-\cos^{-1}x$ is: