
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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  $\triangle ABC$  and 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  $100m+n$ .

*Proposed by David Altizio*

Present the answer in LaTex format: \boxed{Your answer}
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