What is the smallest positive odd integer $n$ such that the product 
\[2^{1/7} \cdot 2^{3/7} \cdot \ldots \cdot 2^{(2n+1)/7}\] 
is greater than $1000$? 
(In the product, the denominators of the exponents are all sevens, and the numerators are the successive odd integers from $1$ to $2n+1$.)

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