
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 $ABC$ and point $P$ in the same plane are given. $\angle APB$ is a right angle, and $\angle APB$ is three times $\angle ACB$. The line segment $\overline{AC}$ intersects $\overline{BP}$ at point $D$. It is known that $PB = 5$ and $PD = 4$. Calculate the product $AD\cdot DC$.
A) 4
B) 7
C) 8
D) 9
E) 10

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