Exponential maximum

Algebra Level 4

$\large { 3 }^{ | x-y | }+{ 3 }^{ | y-z | }+{ 3 }^{ | z-x | } - \sqrt{6x^2+6y^2+6z^2}$

Let $$x, y$$ and $$z$$ be real numbers satisfying $$x + y + z = 0$$, find the minimum value of the expression above.

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