ISL 2023 N6

A sequence of integers $a_{0}, a_{1}, a_{2}, \ldots$ is called kawaii, if $a_{0}=0, a_{1}=1$, and, for any positive integer $n$, we have $$\left(a_{n+1}-3 a_{n}+2 a_{n-1}\right)\left(a_{n+1}-4 a_{n}+3 a_{n-1}\right)=0 .$$ An integer is called kawaii if it belongs to a kawaii sequence. Suppose that two consecutive positive integers $m$ and $m+1$ are both kawaii (not necessarily belonging to the same kawaii sequence). Prove that 3 divides $m$, and that $m / 3$ is kawaii. (China)