\[\large \mathcal{S} = \cos \left(\dfrac{2 \pi}{13} \right) + \cos \left(\dfrac{6 \pi}{13} \right) + \cos \left(\dfrac{8 \pi}{13} \right) \]

If the value of \(\mathcal{S}\) is of the form \(\dfrac{\sqrt{a} - b}{c}\), where \(a,b\) and \(c\) are positive integers with \(a\) square free, find the value of \(a+b+c\).

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