Two circles of radius $5$ are externally tangent to each other and are internally tangent to a circle of radius $13$ at points $A$ and $B$, as shown in the diagram. The distance $AB$ can be written in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $\textbf{(A) } 21 \qquad$ $ \textbf{(B) } 29 \qquad$ $ \textbf{(C) } 58 \qquad$ $ \textbf{(D) } 69 \qquad$ $ \textbf{(E) } 93$