Trevor is at the edge and inside of a circular semicircle of radius 1. He walks a distance of \(d\) towards the radius of the semicircle and climbs up until his head hits the top of the semicircle. Then he walks another distance of \(d\) maintaining the same height as when he hit his head the first time and climbs up again until his head hits the semicircle. This continues like a staircase pattern and when he reaches the top, he stops and shouts in victory. (The GIF shows an illustration of it)

Let \(d=\frac{1}{n}\) where \(n\) is an integer more than 0 and \(D_{n}\) be the total staircase-zigzag distance traveled by Trevor.

Find

\[\lim _{ n\rightarrow \infty }{ D_{n} } \]

Assume Trevor has a height of \(0\)

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