Prove that $a=b=e$

Question

The question is:

Let $(G, *)$ be a group and $a, b \in G$. Suppose that $a*b^3*a^{-1}=b^2$ and $b^{-1}*a^2*b=a^3$. Show that $a=b=e$.

I've tried to prove it in many ways. But, could do nothing so far. Hints are welcome rather than full answers.