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.

(第二十五届AIME1 2007 第11题)