Let $\Gamma_1$ and $\Gamma_2$ be externally tangent circles with radii lengths $2$ and $6$, respectively, and suppose that they are tangent to and lie on the same side of line $\ell$. Points $A$ and $B$ are selected on $\ell$ such that $\Gamma_1$ and $\Gamma_2$ are internally tangent to the circle with diameter $AB$. If $AB = a + b\sqrt{c}$ for positive integers $a, b, c$ with $c$ squarefree, then find $a + b + c$.

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