period of recurrence relation

Question

Find the period of the recurrence sequence with $s_{i+5} = s_{i+1} + s_i$ in $\mathbb{F}_2$ and initial states $s_0 = 1, s_1 = 1, s_2=1, s_3=0, s_4=1$. Firstly, I calculated the characteristic polynomial, $f(z) = z^5 - z - 1$. Then, I tried to find the order of this polynomial, but since it has no roots in $\mathbb{F}_2$, I was not able to do it. Writing down the sequence for some more elements, I assume that the order is 7, but I don't know how to prove it. Can someone please help me?