MathNet 2007-part-02 — 187 problems (EN–VI) Question 142

It is given that $\triangle CAB \cong \triangle EFD$. If $AC = x + y + z$, $AB = z + 6$, $BC = x + 8z$, $EF = 3$, $DF = 2y - z$, and $DE = y + 2$, find $x^{2} + y^{2} + z^{2}$.