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 \(m+n\).

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