Let S be the set of all integer solutions, (x, y, z), of the system of equations
$$
\begin{aligned}
& x-2 y+5 z=0 \\
& -2 x+4 y+z=0 \\
& -7 x+14 y+9 z=0
\end{aligned}
$$
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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