IMC 2024 Problem 7

Let $n$ be a positive integer. Suppose that $A$ and $B$ are invertible $n \times n$ matrices with complex entries such that $A+B=I$ (where $I$ is the identity matrix) and $$(A^2+B^2)(A^4+B^4)=A^5+B^5.$$ Find all possible values of $\det(AB)$ for the given $n$ .