Example 13 (2001 Hunan Province Competition Question) Given $a+b+c=0$, and $a, b, c$ are all non-zero, then simplify $a\left(\frac{1}{b}+\frac{1}{c}\right)+$ $b\left(\frac{1}{a}+\frac{1}{c}\right)+c\left(\frac{1}{a}+\frac{1}{b}\right)$ to $\qquad$ .
```
Solve Fill in 3. Reason: The original expression is decomposed, we get
\[
\begin{aligned}
\text { Original } & =\left(\frac{a}{a}+\frac{b}{a}+\frac{c}{a}\right)+\left(\frac{a}{b}+\frac{b}{b}+\frac{c}{b}\right)+\left(\frac{a}{c}+\frac{b}{c}+\frac{c}{c}\right)-\frac{a}{a}-\frac{b}{b}-\frac{c}{c} \\
& =\frac{0}{a}+\frac{0}{b}+\frac{0}{c}-3=-3
\end{aligned}
\]
```