Example 1. If $a+b+c=0, a b c=0$, find the value of $\frac{a^{2}+b^{2}+c^{2}}{a^{3}+b^{3}+c^{3}}+\frac{2}{3}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)$.
\[\begin{array}{l}\text { Solve the original expression }=\frac{a^{2}+b^{2}+c^{2}}{3 a b c} \\ +\frac{2(a b+b c+c a)}{3 a b c}=\frac{(a+b+c)^{2}}{3 a b c}=0 .\end{array}\]