Order of elements in a group

This probably just a simple manipulation problem but I keep going in circles. It's from Introduction to Algebra by Cameron

A group which contains elements a,b,c,d,e (none equal to the identity) such that ab=c, bc=d, cd=e, de=a, ea=b. Find the orders of a,b,c,d,e.

I found the 'squares' and 'cubes' of the elements, but it starts to get ugly with higher powers. I know I'm probably suppose to be looking for when a pattern repeats for the powers, but I always end up with a horrible expression or back where I started. Any help pointing me in the right direction, please?