\[\large \begin{cases} a^3+ax+y = 0 \\ b^3+bx+y = 0 \\ c^3+cx+y = 0 \end{cases} \]

The system of equations above holds true for some real numbers \(a\), \(b\), \(c\), \(x\), and \(y\), where \(a\), \(b\), and \(c\) are distinct. Find \(a+b+c\).

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