Bernardo chooses a three-digit positive integer $N$ and writes both its base-$2$ and base-$5$ representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-$10$ integers, he adds them to obtain an integer $S$. For how many choices of $N$ are the two rightmost digits of $S$, in order, the same as those of $4N$?