The number of all possible values of $\theta$, where $0<\theta<\pi$, for which the system of equations $$ \begin{aligned} & (y+z) \cos 3 \theta=(x y z) \sin 3 \theta \\ & x \sin 3 \theta=\frac{2 \cos 3 \theta}{y}+\frac{2 \sin 3 \theta}{z} \\ & (x y z) \sin 3 \theta=(y+2 z) \cos 3 \theta+y \sin 3 \theta \end{aligned} $$ have a solution $\left(x_0, y_0, z_0\right)$ with $y_0 z_0 \neq 0$, is