Example 4 - Let $\mu^{2}, \gamma^{2}, \rho^{2}$ be pairwise distinct, and the parameters $\mu, \gamma, \rho, b, c$ be chosen such that the following system of equations has a solution:
$$
\begin{array}{l}
x^{2}>0, y^{2}>0, z^{2}>0, \\
\frac{x^{2}}{\mu^{2}}+\frac{y^{2}}{\mu^{2}-b^{2}}+-\frac{z^{2}}{\mu^{2}-c^{2}}=1, \\
\frac{x^{2}}{\gamma^{2}}+\frac{y^{2}}{\gamma^{2}-b^{2}}+\frac{z^{2}}{\gamma^{2}-c^{2}}=1, \\
\frac{x^{2}}{\rho^{2}}+\frac{y^{2}}{\rho^{2}-b^{2}}+\frac{z^{2}}{\rho^{2}-c^{2}}=1 .
\end{array}
$$

Find the value of $x^{2}+y^{2}+z^{2}$.
Considering the structure characteristics of the known system of equations, we can construct an auxiliary equation about the variable $t$:
$$
\frac{x^{2}}{t}+\frac{y^{2}}{t-b^{2}}+\frac{z^{2}}{t-c^{2}}=1.
$$

Obviously, from the known system of equations, we know that $t=\mu^{2}, \gamma^{2}, \rho^{2}$ are the three roots of equation (1). Furthermore, by eliminating the denominators and rearranging equation (1), we obtain the polynomial equation in $t$:
$$
t^{3}+\alpha t^{2}+\cdots=0,
$$

where $\alpha=-b^{2}-c^{2}-x^{2}-y^{2}-z^{2}$.
Since the three roots of equation (2) are $\mu^{2}, \gamma^{2}, \rho^{2}$, by the generalized Vieta's theorem, we have
$$
\mu^{2}+\gamma^{2}+\rho^{2}=b^{2}+c^{2}+x^{2}+y^{2}+z^{2},
$$

which means
$$
x^{2}+y^{2}+z^{2}=\mu^{2}+\gamma^{2}+\rho^{2}-b^{2}-c^{2}.
$$