Cubic system

Algebra Level 4

\[\begin{cases} 3a = (b+c+d)^3 \\ 3b = (c+d+e)^3 \\ 3c = (d+e+a)^3 \\ 3d = (e+a+b)^3 \\ 3e = (a+b+c)^3 \end{cases}\]

\(a,b,c,d,e\) are real numbers satisfying the above system. Determine the sum of all possible values of \(a\).

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