We select a real number $\alpha$ uniformly and at random from the interval $(0,500)$. Define 
\[ S = \frac{1}{\alpha} \sum_{m=1}^{1000} \sum_{n=m}^{1000} \left\lfloor \frac{m+\alpha}{n} \right\rfloor. \]
Let $p$ denote the probability that $S \ge 1200$. Compute $1000p$.

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