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Let $S$ be the set of all integer solutions, $(x, y, z)$, of the system of equations $x - 2y + 5z = 0$, $-2x + 4y + z = 0$, $-7x + 14y + 9z = 0$ such that $15 \leq x^2 + y^2 + z^2 \leq 150$. Then, the number of elements in the set $S$ is equal to ______.
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