Problem 23 (2003 AMC 8)

In the pattern below, the cat (denoted as a large circle in the figures below) moves clockwise through the four squares and the mouse (denoted as a dot in the figures below) moves counterclockwise through the eight exterior segments of the four squares. If the pattern is continued, where would the cat and mouse be after the $247^{\text{th}}$ move? [See figure in original PDF]