Splendid Series

Geometry Level 5

\[a_n=\cos\left(\frac{\pi}{2n+1}\right) \cos\left(\frac{2\pi}{2n+1}\right) \cos\left(\frac{3\pi}{2n+1}\right)\cdots \cos\left(\frac{(2n+1)\pi}{2n+1}\right)\]

Let \(a_n\) be defined as above for non-negative integer \(n\) and \(S=a_0+a_1+a_2+\cdots \). What is \(\dfrac{1}{1+S}=\boxed{?}\)

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