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

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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