Find the largest possible value of $a + b$ that is less than or equal to $2007$, where $a$ and $b$ are relatively prime, and there exists a positive integer $n$ such that:
\[
\frac{2^3 - 1}{2^3 + 1} \cdot \frac{3^3 - 1}{3^3 + 1} \cdot \frac{4^3 - 1}{4^3 + 1} \cdots \frac{n^3 - 1}{n^3 + 1} = \frac{a}{b}.
\]

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