2024 IMO Shortlist A5

Find all periodic sequences $a_1,a_2,\ldots$ of real numbers such that the following conditions hold for all $n\ge 1$:

$$

a_{n+2}+a_n^2=a_n+a_{n+1}^2

$$

and

$$

|a_{n+1}-a_n|\le 1.

$$

求所有实数周期数列 $a_1,a_2,\ldots$，使得对所有 $n\ge 1$ 都有

$$

a_{n+2}+a_n^2=a_n+a_{n+1}^2

$$

以及

$$

|a_{n+1}-a_n|\le 1。

$$