Can you find the neat solution?

Algebra Level 3

{x26y+17=0y210z+41=0z22x23=0 \begin{cases} x^2-6y + 17 = 0 \\ y ^2 - 10z + 41 = 0 \\ z^2 - 2x- 23 = 0 \end{cases}

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

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