In a certain circle, the chord of a \(d\)-degree arc is 22 centimeters long, and the chord of a \(2d\)-degree arc is 20 centimeters longer than the chord of a \(3d\)-degree arc, where \(d < 120.\) The length of the chord of a \(3d\)-degree arc is \(- m + \sqrt {n}\) centimeters, where \(m\) and \(n\) are positive integers. Find \(m + n.\)

(第十九届AIME1 2001 第13题)