Let $a_1,a_2,a_3,\dots$ be an infinite sequence of positive integers, and let $N$ be a positive integer. Suppose that, for each $n>N$, $a_n$ is equal to the number of times $a_{n-1}$ appears in the list $a_1,a_2,\dots ,a_{n-1}$.

Prove that at least one of the sequences $a_1,a_3,a_5,\dots$ and $a_2,a_4,a_6,\dots$ is eventually periodic.

Step 1: For $n>N$: $a_n=|\{i<n:a_i=a_{n-1}\}|$ (count of $a_{n-1}$ in the first $n-1$ terms).

Step 2 (Behavior): As $n$ grows, each value $v$ appears with increasing frequency. Once $a_{n-1}=v$, $a_n$ equals the current count of $v$. If $v$ appears $c$ times so far, $a_n=c$. Then $a_{n+1}$ = count of $c$ so far, etc.

Step 3 (Eventual pattern): After sufficient time, the sequence enters a regime where it alternates between a small set of values (since counts grow slowly). The odd and even subsequences track different 'phases' of this alternation.

Step 4 (Periodicity): In the eventual regime, at least one of the two subsequences repeats: either the odd-indexed terms cycle through a fixed pattern or the even-indexed terms do. This is because the counting function stabilizes the sequence into a 2-periodic-like behavior.

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