An Interesting Expansion Product

\[\begin{aligned} (x+1)^p(x-3)^q = x^n+a_1x^{n-1}+a_2x^{n-2}+\cdots+a_n \end{aligned}\] Consider the expansion above where \(a_1, a_2, \cdots, a_n, p\) and \(q\) are integers with \(p\) and \(q\) positive and \(n = p+q\). How many ordered pairs \((p, q)\) with \(1\leq p, q < 1000\) are there such that \(a_1 = a_2\)?

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