
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
In $\triangle ABC$, $AB = AC = 13$ and $BC = 10$. $P$ is a point on $BC$ such that $BP < PC$. $H$ and $K$ are the orthocenters of $\triangle APB$ and $\triangle APC$ respectively. If $HK = 2$, find $PC$.


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