BMO Round 1, 2023

A sequence of positive integers $a_n$ begins with $a_1 = a$ and $a_2 = b$ for positive integers $a$ and $b$. Subsequent terms in the sequence satisfy the following two rules for all positive integers $n$:
$$a_{2n+1} = a_{2n}a_{2n-1}, \qquad a_{2n+2} = a_{2n+1} + 4.$$
Exactly $m$ of the numbers $a_1, a_2, a_3, \ldots, a_{2022}$ are square numbers. What is the maximum possible value of $m$? Note that $m$ depends on $a$ and $b$, so the maximum is over all possible choices of $a$ and $b$.