For an odd number $n$, we define $n!! = n \cdot (n-2) \cdot (n-4) \cdots 3 \cdot 1$. Determine how many different residues modulo $1000$ are obtained from $n!!$ when $n = 1, 3, 5, \ldots$.

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