A Harmonic Hex

Calculus Level 2

The image above shows a broken line (a series of connected line segments) starting at the origin, O. The nth 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 \(a+b\).

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