### 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: I notice that the expression involves fractions with absolute values in the denominators, which means that each fraction is either $+1$ or $-1$, depending on the sign of the numerator.
So, I can rewrite the expression as \[\frac{a}{|a|} + \frac{b}{|b|} + \frac{c}{|c|} + \frac{abc}{|abc|} = \text{sign}(a) + \text{sign}(b) + \text{sign}(c) + \text{sign}(abc).\]
Now, I need to think about how the signs of $a$, $b$, and $c$ affect the signs of $abc$ and the whole expression.
If all of $a$, $b$, and $c$ are positive, then $abc$ is also positive, and the expression is $1 + 1 + 1 + 1 = 4$.
If exactly one of $a$, $b$, or $c$ is negative, then $abc$ is also negative, and the expression is $1 + 1 - 1 - 1 = 0$.
If exactly two of $a$, $b$, or $c$ are negative, then $abc$ is positive, and the expression is $1 - 1 - 1 + 1 = 0$.
If all of $a$, $b$, and $c$ are negative, then $abc$ is also negative, and the expression is $-1 - 1 - 1 - 1 = -4$.
So, the only possible values for the expression are $-4$, $0$, and $4$.