Nice Algebra

This is a user suggested problem. The TopsOJ staff thanks and gives full credit to Maximilian113 for their contribution.

Suppose that $a, b, c$ are positive reals such that $a^2+b^2+c^2=18.$ Let $\begin{aligned} x &=\sqrt{a^2+b^2+ab}, \\ y &=\sqrt{b^2+c^2+bc}, \\ z &= \sqrt{c^2+a^2+ca}. \\ \end{aligned}$ Find the maximum possible value of $(x+y+z)(-x+y+z)(x-y+z)(x+y-z).$