
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  $AB=4$ ,  $BC=5$ , and  $CA=6$ . Suppose  $X$  and  $Y$  are points such that



- $BC$  and  $XY$  are parallel
- $BX$  and  $CY$  intersect at a point  $P$  on the circumcircle of  $\triangle{ABC}$
- the circumcircles of  $\triangle{BCX}$  and  $\triangle{BCY}$  are tangent to  $AB$  and  $AC$ , respectively.


Then  $AP^2$  can be written in the form  $\frac{p}{q}$  for relatively prime positive integers  $p$  and  $q$ . Compute  $100p+q$ .

*Proposed by Tristan Shin*

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