
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 $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, respectively. If there exists a point $P$ on the right branch of the hyperbola such that $(\overrightarrow{OP} + \overrightarrow{OF\_2}) \cdot \overrightarrow{F\_2P} = 0$, where $O$ is the coordinate origin, and $|\overrightarrow{PF\_1}| = \sqrt{3}|\overrightarrow{PF\_2}|$, then the eccentricity of the hyperbola is $\_\_\_\_\_\_\_\_$.

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