
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 that the area of $\triangle ABC$ is $4$, $\angle BAC=120^{\circ}$, and point $P$ satisfies $\overrightarrow{BP}=3 \overrightarrow{PC}$, a line passing through point $P$ perpendicular to the lines containing sides $AB$ and $AC$ intersects those lines at points $M$ and $N$ respectively. Find $\overrightarrow{PM} \cdot \overrightarrow{PN}$.

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