
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  $AB=5$ ,  $AC=12$  and incenter  $I$ . Let  $P$  be the intersection of  $AI$  and  $BC$ . Define  $\omega_B$  and  $\omega_C$  to be the circumcircles of  $ABP$  and  $ACP$ , respectively, with centers  $O_B$  and  $O_C$ . If the reflection of  $BC$  over  $AI$  intersects  $\omega_B$  and  $\omega_C$  at  $X$  and  $Y$ , respectively, then  $\frac{O_BO_C}{XY}=\frac{PI}{IA}$ . Compute  $BC$ .

*2016 CCA Math Bonanza Individual #15*

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