Let \Gamma_{1} and \Gamma_{2} be externally tangent circles with radii \frac{1}{2} and \frac{1}{8}, respectively. The line \ell is a common external tangent to \Gamma_{1} and \Gamma_{2}. For n \geq 3, we define \Gamma_{n} as the smallest circle tangent to \Gamma_{n-1}, \Gamma_{n-2}, and \ell. The radius of \Gamma_{10} can be expressed as \frac{a}{b} where a, b are relatively prime positive integers. Find a+b.