Let $ABC$ be a triangle with $AB=20$, $AC=34$, and $BC=42$. Let $\omega_1$ and $\omega_2$ be the semicircles with diameters $\overline{AB}$ and $\overline{AC}$ erected outwards of $\triangle ABC$. Denote by $\ell$ the common external tangent to $\omega_1$ and $\omega_2$. The line through $A$ perpendicular to $\overline{BC}$ intersects $\ell$ at $X$ and $BC$ at $Y$. The length of $\overline{XY}$ can be written in the form $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $100m+n$.

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