For any \( y \in \mathbb{R} \), let \( \cot^{-1}(y) \in (0, \pi) \) and \( \tan^{-1}(y) \in \left( -\frac{\pi}{2}, \frac{\pi}{2} \right) \). Then the sum of all the solutions of the equation \( \tan^{-1}\!\left( \frac{6y}{9-y^{2}} \right) + \cot^{-1}\!\left( \frac{9-y^{2}}{6y} \right) = \frac{2\pi}{3} \) for \( 0 < |y| < 3 \), is equal to:
