Let \(N = \sum_{k = 1}^{1000} k ( \lceil \log_{\sqrt{2}} k \rceil - \lfloor \log_{\sqrt{2}} k \rfloor )\)
Find the remainder when \(N\) is divided by 1000. (\(\lfloor{k}\rfloor\) is the greatest integer less than or equal to \(k\), and \(\lceil{k}\rceil\) is the least integer greater than or equal to \(k\).)
(第二十五届AIME1 2007 第7题)