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Geometry Level 4

The number of all possible values of \(\theta\), where \(0<\theta<\pi\), for which the system of equations: \[ (y+z)\cos 3\theta=(xyz)\sin 3\theta \\ x\sin 3\theta=\dfrac{2\cos 3\theta}{y}+\dfrac{2\sin 3\theta }{z}\\ (xyz)\sin 3\theta=(y+2z)\cos 3\theta +y\sin 3\theta\] have a solution \((x_0,y_0,z_0)\) with \(y_0z_0\neq0\), is:

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