A short way and a short way

Algebra Level 5

$\large{ \begin{cases} z^x = x \\ z^y = y \\ y^y = x \end{cases} }$

Find all triplets of positive numbers $$(x,y,z)$$ satisfying the system of equations above. Submit your answer as $$\displaystyle \sum (x_k^2 + y_k^2 + z_k^2 )$$.

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