Question 13
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 $d \lt 120.$ The length of the chord of a $3d$-degree arc is $-m + \sqrt{n}$ centimeters, where $m$ and $n$ are positive integers. Find $m + n.$