Newton's Sums with euler's number?

Algebra Level 3

\[x+y+z=0\\ { x }^{ 2 }+{ y }^{ 2 }+{ z }^{ 2 }=e\\ { x }^{ 3 }+{ y }^{ 3 }+{ z }^{ 3 }={ e }^{ 2 }\\ \]

This set of equations is true for three complex numbers \(x, y, z\), where \(e\) is Euler's Constant.

If \(xyz=\frac { A }{ B } { e }^{ C }\) for positive integers \(A,B,C\) with coprime \(A,B\), find \(A+B+C\).

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