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 of one of the circular faces of the cylinder so that $\widehat{\mathrm{AB}}$ 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 these 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 $a+b+c$ . 

Let's think step by step and output the final answer within \boxed{}.
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