Let $x = \sin (2{\tan ^{ - 1}}\alpha )$ and $y = \sin \left( {{1 \over 2}{{\tan }^{ - 1}}{4 \over 3}} \right)$. If $S = \{ a \in R:{y^2} = 1 - x\} $, then $\sum\limits_{\alpha \in S}^{} {16{\alpha ^3}} $ is equal to _______________.