
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
Points \( A, B, C, \) and \( D \) are consecutively placed along a straight line with \( AB = BC = CD \). A point \( P \) is positioned such that \( \cos \angle APC = \frac{3}{5} \) and \( \cos \angle BPD = \frac{12}{13} \). Find \( \cos (2 \angle BPC) \).

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