Question 2 Given that $a, b, c$ are non-zero real numbers, and
$$
M=\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{a b}{|a b|}+\frac{b c}{|b c|}+\frac{c a}{|c a|}+\frac{a b c}{|a b c|} \text {. }
$$

Find the value of $M$.
Solution ${ }^{[2]}$ Considering the "equal status" of $a, b, c$ in $M$, the value of $\frac{a b}{|a b|}+\frac{b c}{|b c|}+\frac{c a}{|c a|}$ depends on the sign of the product of two numbers, and the value of $\frac{a b c}{|a b c|}$ depends on the sign of the product of three numbers. Therefore, we can classify according to the number of positive numbers among $a, b, c$:
(1) When $a, b, c$ are all positive, $M=7$.
(2) When $a, b, c$ have two positive numbers and one negative number, $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}=1$,
$$
\frac{a b}{|a b|}+\frac{b c}{|b c|}+\frac{c a}{|c a|}=-1, \frac{a b c}{|a b c|}=-1 \text {. }
$$

Thus, $M=-1$.
(3) When $a, b, c$ have one positive number and two negative numbers,
$$
\begin{array}{l}
\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}=-1, \\
\frac{a b}{|a b|}+\frac{b c}{|b c|}+\frac{c a}{|c a|}=-1, \frac{a b c}{|a b c|}=1 .
\end{array}
$$

Thus, $M=-1$.
(4) When $a, b, c$ are all negative, $M=-1$.

In summary, the value of $M$ is -1 or 7.