Problem 9 (2010 AIME I)

Let $(a, b, c)$ be a real solution of the system of equations \begin{align*} x^3 - xyz &= 2 \ y^3 - xyz &= 6 \ z^3 - xyz &= 20 \end{align*} The greatest possible value of $a^3 + b^3 + c^3$ can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.