2023 IMO Shortlist 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)

整数序列 $a_{0}, a_{1}, a_{2}, \ldots$ 被称为卡哇伊，如果 $a_{0}=0, a_{1}=1$，并且对于任何正整数 $n$，我们有 $$\left(a_{n+1}-3 a_{n}+2 a_{n-1}\right)\left(a_{n+1}-4 a_{n}+3 a_{n-1}\right)=0 。$$ 如果一个整数属于卡哇伊序列，则该整数被称为卡哇伊。假设两个连续的正整数$m$和$m+1$都是kawaii(不一定属于同一个kawaii序列)。证明 3 整除 $m$，并且 $m / 3$ 很卡哇伊。 (中国)