<image>As shown in diagram ①, the side length of the large square is \(3a-7\) and the side length of the small square in the middle is \(a+5\). As shown in diagram ②, the width of the rectangle is \(4(a-6)\). If the area of the shaded region in diagram ① is equal to the area of the rectangle in diagram ②, find the length of the rectangle.
