Polynomial Again?

Algebra Level 5

$\begin{cases} x^3-xyz=2 \\ y^3-xyz =6 \\ z^3-xyz = 20 \end{cases}$

If $x,y$ and $z$ are real solutions satisfying the system of equations above, and the maximum value of $x^3+y^3+z^3$ is equal to $\dfrac mn$ , where $m$ and $n$ are coprime positive integers, find $m+n$ .