Can you find the neat solution?

Algebra Level 3

\[ \begin{cases} x^2-6y + 17 = 0 \\ y ^2 - 10z + 41 = 0 \\ z^2 - 2x- 23 = 0 \end{cases} \]

If \(x,y\) and \(z\) are real numbers that satisfy the system of equations above, find the sum of all distinct value(s) of \(x+y+z\).

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