Suppose $a$, $b$, $c$, and $d$ are nonzero real numbers, and $a+b+c+d=0$. What are the possible value(s) for $\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{d}{|d|}+\frac{abcd}{|abcd|}$?