sequences and series - Difficult nonlinear system based on max value

Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 - xyz = 2$, $ y^3 - xyz = 6$, $ z^3 - xyz = 20$. 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$.

I got that:

$$a^3 + b^3 + c^3 = 28 + 3abc$$

I tried using a substitution,

$t = abc$ to get:

$a^3 + b^3 + c^3 = 28 + 3t$ but I cannot replace the LHS.

Can somebody just help me with the substitution, thats all!?

Answer

Hint.

You have $$\left\{
\begin{aligned}
x^3=2+t\\
y^3=6+t\\
z^3=20+t
\end{aligned}
\right.$$ Hence $$x^3 y^3 z^3 = t^3 = (2+t)(6+t)(20+t)$$