
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
In  $\triangle ABC$ , let  $X$  and  $Y$  be points on segment  $BC$  such that  $AX=XB=20$  and  $AY=YC=21$ . Let  $J$  be the  $A$ -excenter of triangle  $\triangle AXY$ . Given that  $J$  lies on the circumcircle of  $\triangle ABC$ , the length of  $BC$  can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $m+n$ .

*Proposed by Andrew Wen*

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