
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 the hyperbola $x^{2} - \frac{y^{2}}{3} = 1$ with its left and right foci denoted as $F_1$ and $F_2$ respectively and eccentricity $e$, if a point $P$ on the hyperbola satisfies $\frac{\sin \angle PF_2F_1}{\sin \angle PF_1F_2} = e$, determine the value of $\overrightarrow{F_2P} \cdot \overrightarrow{F_2F_1}$.

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