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 _______________.

$$50\tan \left( {3{{\tan }^{ - 1}}\left( {{1 \over 2}} \right) + 2{{\cos }^{ - 1}}\left( {{1 \over {\sqrt 5 }}} \right)} \right) + 4\sqrt 2 \tan \left( {{1 \over 2}{{\tan }^{ - 1}}(2\sqrt 2 )} \right)$$ is equal to ____________.