Let
N=\sum_{k=1}^{1000} k\left(\left\lceil\log _{\sqrt{2}} k\right\rceil-\left\lfloor\log _{\sqrt{2}} k\right\rfloor\right)
Find the remainder when N is divided by 1000. (Here \lfloor x\rfloor denotes the greatest integer that is less than or equal to x, and \lceil x\rceil denotes the least integer that is greater than or equal to x.)