AMO 2020 Problem 5

Each term of an infinite sequene $a_1,a_2, \cdots$ is equal to 0 or 1. For each positive integer $n$ , [list] [] $a_n+a_{n+1} \neq a_{n+2} +a_{n+3}$ and [] $a_n + a_{n+1}+a_{n+2} \neq a_{n+3} +a_{n+4} + a_{n+5}$ Prove that if $a_1~=~0$ , then $a_{2020}~=~1$