Let $p$ be prime and $K$ a field with $char(K)=p$. Let $A \in M_n(K)$ such that $A^p=I$. Find the Jordan-Chevalley decomposition of A

Question

Let $p$ be prime and $K$ a field with $char(K)=p$. Let $A \in M_n(K)$ such that $A^p=I$. Find the Jordan-Chevalley. I got a hint which says, that I should write $A^p-I$ as a power of some other matrix. However, I was not able to find it. I thought that maybe the formula for roots of unity might help i.e. $I+A+A^2+\dots +A^{p-1}$ but I did not find a way to go on. Do someone has a tip for me?