
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 equally spaced along a line such that \(AB = BC = CD = 1.\) A point \(P\) is located such that \(\cos \angle APC = \frac{3}{5}\) and \(\cos \angle BPD = \frac{1}{5}.\) Determine \(\sin (2 \angle BPC).\)

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