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Calculus Level 3

\[\sum_{n=1}^\infty \binom{4n}{2n}^{-1}=X+\frac{\pi}{\sqrt{Y}}-\frac{2}{Z\sqrt{Z}}\ln\left(\frac{1+\sqrt{Z}}{2}\right)\]

The equation above holds true for rational numbers \(X\), \(Y\), and \(Z\). Find \(\sqrt{XYZ}\).

Note: \(\binom{\cdot}{\cdot}\) is the binomial coefficient. The first few terms of the series are as follows:

\[\begin{align} \sum_{n=1}^\infty \binom{4n}{2n}^{-1} &= \frac{1}{\binom{4}{2}} + \frac{1}{\binom{8}{4}} + \frac{1}{\binom{12}{6}} + \cdots \\ &= \frac{1}{6}+\frac{1}{70}+\frac{1}{924}+\cdots. \end{align}\]

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