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 ____________.
