Example 6. (USAMO problem) Solve the system of equations
$$
\left\{\begin{array}{l}
x+y+z=3, \\
x^{2}+y^{2}+z^{2}=3, \\
x^{5}+y^{5}+z^{5}=3,
\end{array}\right.
$$

Find all real or complex roots.
Let $x, y, z$ be the roots of the cubic equation $r^{3}-a r^{2}+b r -c=0$. Then, by (1) and (2), we know
$$
x y+y z+z x=3 \text {, }
$$

Therefore, the cubic equation is
$$
r^{3}-3 r^{2}+3 r-c=0 \text {. }
$$

Let $c=1+m^{3}$ (where $m$ is a complex number), then $(r-1)^{3} = m^{3}$. Thus, $r=1+m, 1+\omega m, 1+\omega^{2} m$, where $\omega=-\frac{1}{2}+\frac{\sqrt{3}}{2} i$.

By (3), we have $(1+m)^{5}+(1+\omega m)^{5}+(1+\omega^{2} m)^{5}=3$. Noting that
$1+\omega^{n}+\omega^{2 n}=\left\{\begin{array}{ll}0, & n \in\{3 k+1\} \text { or }\{3 k+2\} \\ 3, & n \in\{3 k\} .\end{array}\right.$, we get $30 m^{3}=0$, i.e., $m=0$.
Therefore, all the roots are $x=y=z=1$.