EGMO 2020 Problem 6

Let $m > 1$ be an integer. A sequence $a_1, a_2, a_3, \ldots$ is defined by $a_1 = a_2 = 1$ , $a_3 = 4$ , and for all $n \ge 4$ , $$a_n = m(a_{n - 1} + a_{n - 2}) - a_{n - 3}.$$ Determine all integers $m$ such that every term of the sequence is a square.