Example 11 Let $x, y, z, A, B, C \in \mathbf{R}$, and
$$
A+B+C=\pi \text {. }
$$

Let $G_{n}=x^{n} \sin n A+y^{n} \sin n B+z^{n} \sin n C\left(n \in \mathbf{Z}_{+}\right)$.
If $G_{1}=G_{2}=0$, try to find the value of $G_{n}$.