A leash of length \(L\) is secured to the side of a circular pen of radius \(r = 100\text{ m}\) so that the region accessible from the free end of the leash is half of the area of the pen.

What is the length \(L\) of the leash to the nearest meter?

Assume \(100 \text{ m} < L < 141 \text{ m} \).

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