It gives practice in working systematically to consider all cases. First however you need to cut down the number of cases to be checked, so think about the relation between $a$, $b$ and $c$ and what you can deduce from it about the constraints on $a$, $b$ and $c$.
Ask the learners to find a relation between $a$, $b$ and $c$. Then have a class discussion about what can be deduced from this expression.
What do you know about $c^3-1$?