Let $x, y, z$ be complex numbers satisfying the following system of equations:  
\[  
\begin{align*}  
z^2 + 5x &= 10z, \\  
y^2 + 5z &= 10y, \\  
x^2 + 5y &= 10x.  
\end{align*}  
\]  
Find the sum of all possible values of $z$.

Let's think step by step and output the final answer within \boxed{}.
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