Example 1 Solve the system of equations
$$
\left\{\begin{array}{l}
x+a y+a^{2} z+a^{3}=0 \\
x+b y+b^{2} z+b^{3}=0 \\
x+c y+c^{2} z+c^{3}=0
\end{array}\right.
$$
Analysis: Observing the shape characteristics of the three equations in the system, it inspires the construction of a cubic equation
$$
t^{3}+z t^{2}+y t+x=0,
$$

where $a, b, c$ are its three roots, and $a, b, c$ are also the roots of the equation
$$
(t-a)(t-b)(t-c)=0.
$$

Expanding the latter equation yields
$$
t^{3}-(a+b+c) t^{2}+(a b+b c+c a) t-a b c=0.
$$

Comparing the coefficients of these two cubic equations, we find that the solutions to the original system of equations are $x=-a b c, y=a b+b c+c a, z=-(a+b+c)$.