### 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: To solve this problem, I need to consider the signs of $a$, $b$, and $c$.
If $a$, $b$, and $c$ are all positive, then the expression becomes $1 + 1 + 1 + 1 = 4$.
If $a$, $b$, and $c$ are all negative, then the expression becomes $-1 - 1 - 1 - 1 = -4$.
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 $-1 - 1 + 1 + 1 = 0$.
I notice that the expression is always an integer, and it is either $4$, $-4$, or $0$.
Therefore, the only possible values of the expression are $4$, $-4$, and $0$.