\[ \dfrac{1}{1-2^{2}}+\dfrac{1}{1-4^{2}}+\dfrac{1}{1-6^{2}}+\dfrac{1}{1-8^{2}}+\cdots+\dfrac{1}{1-1000^{2}}\ \]

The value of the expression above can be expressed as \(\dfrac{a}{b},\) where \(a\) and \(b\) are coprime integers.

What is the number of ways (equivalently, the number of ordered pair of integers \((a,b)\) ) the value of the expression above can be expressed as \(\dfrac{a}{b},\) where \(a\) and \(b\) are coprime integers.

**Hint**: Is it at all needed to compute the sum?

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