Example 2. Given that 1 is a root of the equation $a x^{2}+b x+c=0$, find the value of $\frac{a^{2}+b^{2}+c^{2}}{a^{3}+b^{3}+c^{3}}+\frac{2}{3}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$.
Given that $1$ is a root of the equation $a x^{2}+b x+c=0$,
$$
\begin{array}{c}
\therefore a+b+c=0 . \\
\text { Also, } a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c) \\
.\left(a^{2}+b^{2}+c^{2}-a b-b c-c a\right) .
\end{array}
$$

From (1) and (2), we can deduce
$$
\begin{array}{c}
a^{3}+b^{3}+c^{3}=3 a b c . \\
\text { Therefore, the original expression }=\frac{a^{2}+b^{2}+c^{2}}{3 a b c} \\
+\frac{2(a b+b c+c a)}{3 a b c} \\
=\frac{(a+b+c)^{2}}{3 a b c}=0 .
\end{array}
$$