imc

Let the center of the surrounding circle be $X$. The circle that is tangent at point $A$ will have point $Y$ as the center. Similarly, the circle that is tangent at point $B$ will have point $Z$ as the center. Connect $AB$, $YZ$, $XA$, and $XB$. Now observe that $△XYZ$ is similar to $△XAB$ by SAS. Writing out the ratios, we get $$\frac{XY}{XA}=\frac{YZ}{AB}\Rightarrow \frac{13-5}{13}=\frac{5+5}{AB}\Rightarrow \frac{8}{13}=\frac{10}{AB}\Rightarrow AB=\frac{65}{4}.$$ Therefore, our answer is $65+4=\boxed{69}$.