Example 1 Given $a b c=1$, find the value of $\frac{a}{a b+a+1}+\frac{b}{b c+b+1}+\frac{c}{c a+c+1}$.
Replace 1 with $a b c$ in the problem, we get
$$
\text { original expression }=\frac{a}{a b+a+a b c}+\frac{b}{b c+b+1}+\frac{b c}{b c+b+1}=\frac{b c+b+1}{b c+b+1}=1 \text {. }
$$