A sequence with the first term $a_0$ is defined such that $a_{n+1} = 2a_n^2 - 1$ for $n \geq 0$. Let $N$ denote the number of possible values of $a_0$ such that $a_0 = a_{2020}$. Find the number of factors of $N$.

Let's think step by step and output the final answer within \boxed{}.
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