
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  $AB=13$ ,  $BC=14$ , and  $CA=15$ . The incircle of  $ABC$  meets  $BC$  at  $D$ . Line  $AD$  meets the circle through  $B$ ,  $D$ , and the reflection of  $C$  over  $AD$  at a point  $P\neq D$ . Compute  $AP$ .

*2020 CCA Math Bonanza Tiebreaker Round #4*

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