For how many positive integers $n$, $1 \leq n \leq 2008$, can the set $\{1, 2, 3, \ldots, 4n\}$ be divided into $n$ disjoint 4-element subsets such that every one of the $n$ subsets contains the element which is the arithmetic mean of all the elements in that subset?

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