An algebra problem by Sompong Chuisurichy

Algebra Level 4

Let \(a,b,c\) be real numbers such that \(0<|a|<|b|<|c|\). How many possible values of

\( \dfrac{a}{|a|} + \dfrac{b}{|b|} + \dfrac{c}{|c|} + \dfrac{a+b}{|a+b|}+\dfrac{b+c}{|b+c|}+\dfrac{c+a}{|c+a|}? \)

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