### Instruction: Given that $a$, $b$, and $c$ are nonzero real numbers, find all possible values of the expression
\[\frac{a}{|a|} + \frac{b}{|b|} + \frac{c}{|c|} + \frac{abc}{|abc|}.\]Enter all possible values, separated by commas.### Response: This expression looks like it can only take a few discrete values, depending on the signs of $a$, $b$, and $c$.
Let's see what happens in each case.
If all three of $a$, $b$, and $c$ are positive, then the absolute value bars do nothing, and the expression simplifies to $1 + 1 + 1 + 1 = 4$.
If exactly one of $a$, $b$, and $c$ is negative, say $a$, then the expression becomes $-1 + 1 + 1 - 1 = 0$.
If exactly two of $a$, $b$, and $c$ are negative, say $a$ and $b$, then the expression becomes $-1 - 1 + 1 + 1 = 0$ as well.
If all three of $a$, $b$, and $c$ are negative, then the expression becomes $-1 - 1 - 1 - 1 = -4$.
So, the only possible values of the expression are $-4$, $0$, and $4$.