
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  $AB = 20$  and  $AC = 22$ . Suppose its incircle touches  $\overline{BC}$ ,  $\overline{CA}$ , and  $\overline{AB}$  at  $D$ ,  $E$ , and  $F$  respectively, and  $P$  is the foot of the perpendicular from  $D$  to  $\overline{EF}$ . If  $\angle BPC = 90^{\circ}$ , then compute  $BC^2$ .

*Proposed by Ankan Bhattacharya*

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