Why do this problem?
To beat the spies at their own game!

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 $c$.

Possible approach
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.

Key question
What do you know about $c^3-1$?