Example 3 Given $a b c<0$, let
$$
P=\frac{a}{|a|}+\frac{|b|}{b}+\frac{c}{|c|}+\frac{|a b|}{a b}+\frac{a c}{|a c|}+\frac{|b c|}{b c} \text {. }
$$

Find the value of $a P^{3}+b P^{2}+c P+2009$.
Given $a b c<0$, we get $\frac{a b c}{|a b c|}=-1$, and at least one of $a, b, c$ is negative. According to the corollary, we can change the position of the absolute value symbol, then
$$
\begin{aligned}
P & =\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{a b}{|a b|}+\frac{a c}{|a c|}+\frac{b c}{|b c|} \\
& =\left(1+\frac{a}{|a|}\right)\left(1+\frac{b}{|b|}\right)\left(1+\frac{c}{|c|}\right)-1-\frac{a b c}{|a b c|} \\
& =0-1+1=0 .
\end{aligned}
$$

Therefore, $a P^{3}+b P^{2}+c P+2009=2009$.