
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 $ABC$ is a right triangle with $\angle BAC = 90^\circ$, where $AB = 12$, $BC = 5$, and $CA = 13$. Point $P$ is inside triangle $ABC$ where angles $PAB$, $PBC$, and $PCA$ are all congruent. Calculate $\cos \angle PAB$.

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