Let $a_0,b_0,c_0$ be complex numbers, and define
$$a_{n+1}=a_n^2+2b_nc_n$$
$$b_{n+1}=b_n^2+2c_na_n$$
$$c_{n+1}=c_n^2+2a_nb_n$$
for all nonnegative integers $n$.
Suppose that $\max{|a_n|,|b_n|,|c_n|}\leq2022$ for all $n$. Prove that
$$|a_0|^2+|b_0|^2+|c_0|^2\leq 1.$$