$\begin{cases} x + 7y + 3v + 5u = 16 \\ 8x + 4y + 6v + 2u = -16 \\ 2x + 6y + 4v + 8u = 16 \\ 5x + 3y + 7v + u = -16 \end{cases}$

Real numbers $x$, $y$, $u$, and $v$ satisfy the system of equations above. Find $xyuv$.

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