\[\large \sum_{n=1}^\infty \frac{1}{{2n \choose n}} \]

Suppose the following infinite sum can be written as \[ \dfrac{a+\pi b\sqrt{c}}{d},\]

where \(a,b,c\) and \(d\) are integers with \(b\) and \(c\) prime. Then compute the value of \(a+b+c+d\),

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