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 $$x^3 y^3 z^3 = t^3 = (2+t)(6+t)(20+t)$$