For each positive integer $p$, let $b(p)$ denote the unique positive integer $k$ such that $|k-\sqrt{p}| < \frac{1}{2}$.  For example, $b(6) = 2$ and $b(23) = 5$.  If $S = \sum_{p=1}^{2007} b(p),$ find the remainder when $S$ is divided by 1000.


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