A block of wood has the shape of a right circular cylinder with radius \(6\) and height \(8\), and its entire surface has been painted blue. Points \(A\) and \(B\) are chosen on the edge on one of the circular faces of the cylinder so that \(\widehat{AB}\) on that face measures \(120^\circ\). The block is then sliced in half along the plane that passes through point \(A\), point \(B\), and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of those unpainted faces is \(a\cdot\pi + b\sqrt{c}\), where \(a\), \(b\), and \(c\) are integers and \(c\) is not divisible by the square of any prime. Find \(a+b+c\).

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