
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  $AB = 7, BC = 5,$  and  $CA = 6$ . Let  $D$  be a variable point on segment  $BC$ , and let the perpendicular bisector of  $AD$  meet segments  $AC, AB$  at  $E, F,$  respectively. It is given that there is a point  $P$  inside  $\triangle ABC$  such that  $\frac{AP}{PC} = \frac{AE}{EC}$  and  $\frac{AP}{PB} = \frac{AF}{FB}$ . The length of the path traced by  $P$  as  $D$  varies along segment  $BC$  can be expressed as  $\sqrt{\frac{m}{n}}\sin^{-1}\left(\sqrt \frac 17\right)$ , where  $m$  and  $n$  are relatively prime positive integers, and angles are measured in radians. Compute  $100m + n$ .

*Proposed by Edward Wan*

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