Example 3 Given that $a, b, c$ are three distinct real numbers. Try to solve the system of equations about $x, y, z$
$$
\left\{\begin{array}{l}
\frac{x}{a^{3}}-\frac{y}{a^{2}}+\frac{z}{a}=1, \\
\frac{x}{b^{3}}-\frac{y}{b^{2}}+\frac{z}{b}=1, \\
\frac{x}{c^{3}}-\frac{y}{c^{2}}+\frac{z}{c}=1 .
\end{array}\right.
$$
【Analysis】This is a classic problem involving the application of Vieta's formulas to a univariate cubic equation. Solving the system of equations directly would involve a large amount of computation. Observing the characteristics of the three equations, we know that $a$, $b$, and $c$ are the roots of the equation $\frac{x}{m^{3}}-\frac{y}{m^{2}}+\frac{z}{m}=1$ with respect to $m$. This equation can be transformed into
$$
m^{3}-z m^{2}+y m-x=0 \text {. }
$$

According to Vieta's formulas, we have
$$
x=a b c, y=a b+b c+c a, z=a+b+c \text {. }
$$

This is the solution to the problem.