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We can write $$\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{abc}{|abc|}=\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{a}{|a|}\cdot \frac{b}{|b|}\cdot \frac{c}{|c|}.$$Note that $\frac{a}{|a|}$ is 1 if $a$ is positive, and $-1$ if $a$ is negative. Thus, $\frac{a}{|a|}$ depends only on the sign of $a$, and similarly for the terms $\frac{b}{|b|}$ and $\frac{c}{|c|}$.

Furthermore, the expression is symmetric in $a$, $b$, and $c$, so if $k$ is the number of numbers among $a$, $b$, and $c$ that are positive, then the value of the given expression depends only on $k$.

If $k=3$, then $$\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{a}{|a|}\cdot \frac{b}{|b|}\cdot \frac{c}{|c|}=1+1+1+1\cdot 1\cdot 1=4.$$If $k=2$, then $$\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{a}{|a|}\cdot \frac{b}{|b|}\cdot \frac{c}{|c|}=1+1+(-1)+1\cdot 1\cdot (-1)=0.$$If $k=1$, then $$\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{a}{|a|}\cdot \frac{b}{|b|}\cdot \frac{c}{|c|}=1+(-1)+(-1)+1\cdot (-1)\cdot (-1)=0.$$If $k=0$, then $$\frac{a}{|a|}+\frac{b}{|b|}+\frac{c}{|c|}+\frac{a}{|a|}\cdot \frac{b}{|b|}\cdot \frac{c}{|c|}=(-1)+(-1)+(-1)+(-1)\cdot (-1)\cdot (-1)=-4.$$Therefore, the possible values of the expression are $\boxed{4,0,-4}$.