$A, B, C$ are points in the plane such that $\angle ABC = 90^\circ$. Circles with diameters $BA$ and $BC$ meet at $D$. If $BA = 20$ and $BC = 21$, then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?

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