USAJMO 2022 Problem 6

Let $a_0, b_0, c_0$ be complex numbers, and define \begin{align*}a_{n+1} &= a_n^2 + 2b_nc_n b_{n+1} &= b_n^2 + 2c_na_n c_{n+1} &= c_n^2 + 2a_nb_n\end{align*}for all nonnegative integers $n.$

Suppose that $\max{\{|a_n|, |b_n|, |c_n|\}} \leq 2022$ for all $n.$ Prove that $$|a_0|^2 + |b_0|^2 + |c_0|^2 \leq 1.$$