Let \( x = \sin\!\left(2 \tan^{-1}\!\alpha\right) \) and \( y = \sin\!\left(\dfrac{1}{2}\tan^{-1}\!\dfrac{4}{3}\right) \). If \( S = \{ \alpha \in \mathbb{R} : y^2 = 1 - x \} \); then \( \displaystyle\sum_{\alpha \in S} 16\alpha^3 \) is equal to __________.
