A sequence $\{a_n\}$ is defined such that $a_i = i$ for $i = 1, 2, 3, \ldots, 2020$ and for $i > 2020$, $a_i$ is the average of the previous 2020 terms. What is the largest integer less than or equal to $\displaystyle\lim_{n \to \infty} a_n$?

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