I'm invoking the Fractional-Part Function!

\[\large{ \{q\}^2 = \left \{ \dfrac{n!}{2000} \right \} }\]

How many pairs \((n,q)\) satisfy the above equation, where \(n\) is a positive integer and \(q\) is a non-integer rational number such that \( 0<q<2000\)?

Notation: \(\{r\}\) denotes the fractional part function.

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