$\large S = \frac{1^{2}}{1\times3} + \frac{2^{2}}{3\times5} + \frac{3^{2}}{5\times7} + \cdots + \frac{500^{2}}{999\times1001}$

For $S$ as defined above, find the number of divisors of $\left \lfloor S \right \rfloor$.

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**Notation:** $\lfloor \cdot \rfloor$ denotes the floor function.