Scary equations

\[\begin{cases} x^2 + 2y^2 + 5z^2 = 35 + 6yz \\ x + 2y - 3z = 9 \\ x + y + z = 10 \end{cases}\]

Given that the integers \((x, y, z)\) are a solution to the following simultaneous equations above, find the value of \(xyz\).

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