A geometry problem by Rocco Dalto

Geometry Level 4

\[ \begin{align} \cos \dfrac \pi A & =\dfrac12 \sqrt{2 + \sqrt{2 + \sqrt3}} \\ \cos \dfrac \pi B & =\dfrac12 \sqrt{2 + \sqrt{2 + \sqrt2}} \\ \cos \dfrac \pi C & =\dfrac12 \sqrt{2 + \sqrt{2 + \sqrt1}} \end{align} \]

If \(A\), \(B\) and \(C\) are positive numbers satisfying the system of equations above, find \(A + B + C.\)

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