Middle European Mathematical Olympiad 2022 Problem T-1

Given a pair $(a_0, b_0)$ of real numbers, we define two sequences $a_0, a_1, a_2, \ldots$ and $b_0, b_1, b_2, \ldots$ of real numbers by $$a_{n+1} = a_n + b_n \quad \text{and} \quad b_{n+1} = a_n \cdot b_n$$ for all $n = 0, 1, 2, \ldots$. Find all pairs $(a_0, b_0)$ of real numbers such that $a_{2022} = a_0$ and $b_{2022} = b_0$.