Let $a = \pi/2008$. Find the smallest positive integer $n$ such that  
$$2[\cos(a)\sin(a) + \cos(4a)\sin(2a) + \cos(9a)\sin(3a) + \cdots + \cos(n^2a)\sin(na)]$$  
is an integer.

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