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)