Mathematics Question from JEE Main 2023 · 25 January, Shift 1

If the sum of all the solutions of $\tan^{-1}\left(\frac{2x}{1-x^2}\right) + \cot^{-1}\left(\frac{1-x^2}{2x}\right) = \frac{\pi}{3}$, where $-1 < x < 1, x \neq 0$, is $\alpha - \frac{4}{\sqrt{3}}$, then $\alpha$ is equal to _____.