*broken line* (a series of connected line segments) starting at the origin, *O*. The *n*th segment in the broken line has length $\frac{1}{n}$, and at the end of each segment, the broken line turns $60^{\circ}$ counter-clockwise.

As the number of segments in the broken line approaches infinity, the final endpoint of the broken line approaches a point *P*. The distance *OP* can be written as $\frac{a}{b}\pi$, where *a* and *b* are positive coprime integers. Find $a+b$.