
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
Triangle  $ABC$  has  $\overline{AB} = \overline{AC} = 20$  and  $\overline{BC} = 15$ . Let  $D$  be the point in  $\triangle ABC$  such that  $\triangle ADB \sim \triangle BDC$ . Let  $l$  be a line through  $A$  and let  $BD$  and  $CD$  intersect  $l$  at  $P$  and  $Q$ , respectively. Let the circumcircles of  $\triangle BDQ$  and  $\triangle CDP$  intersect at  $X$ . The area of the locus of  $X$  as  $l$  varies can be expressed in the form  $\tfrac{p}{q}\pi$  for positive coprime integers  $p$  and  $q$ . What is  $p + q$ ?


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