Define a sequence recursively by $F_0=0$, $F_1=1$, and $F_n=$ the remainder when $F_{n-1}+F_{n-2}$ is divided by $4$, for all $n\ge 2$. What is $F_{2020}+F_{2021}+F_{2022}+F_{2023}+F_{2024}+F_{2025}$?