Let $x=\sin \left(2 \tan ^{-1} \alpha\right)$ and $y=\sin \left(\frac{1}{2} \tan ^{-1} \frac{4}{3}\right)$. If $S=\left\{\alpha \in \mathbb{Z}: y^2=1-x\right\}$, then $\sum_{\alpha=8} 16 \alpha^3$ is equal to $\_\_\_\_$ .
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