The Symmetric Product Challenge

Let $x$, $y$, $z>0$ satisfy$$ \begin{cases} x + y + z = 6\\ x^2+y^2+z^2=14 \end{cases} $$Find the maximum possible value of$$xyz(x-y)^2(y-z)^2(z-x)^2$$The answer is in the form $\displaystyle \frac{a\sqrt b}{c}-d$ where $a$, $b$, $c$, $d$ are positive integers and $a$, $b$, $c$ are coprime. Input the value of $a+b+c+d$.