
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
Given triangle $ABC$ and point $P$ in the same plane, point $P$ is equidistant from $A$ and $B$, angle $APB$ is three times angle $ACB$, and $\overline{AC}$ intersects $\overline{BP}$ at point $D$. If $PB = 4$ and $PD = 1$, calculate $AD\cdot CD$.

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