Periods of an LFSR with characteristic polynomial that is a product of primitive polynomials

Question

I want to find the minimal period of any state of an LFSR (except the initial state of all zeroes) whose characteristic polynomial is the product of two primitive polynomials.

In particular, $f(x),g(x) \in GF(2)[x]$ are primitive polynomials of order $n$, and the characteristic polynomial of the LFSR is given by: $$c(x)=f(x)g(x).$$ What is the minimal period of any state of this LFSR, other than the all-zeros state?

I've tried to say that if $c(x)$ is a product of two primitives then it has a period of $$\pi=2^{2n}-1 $$ but my math gets me to $$\pi =2^{n+1}-2$$ What did I do wrong?