
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
Let \( F_{1} \) and \( F_{2} \) be the left and right foci of the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (where \( a > 0 \) and \( b > 0 \)). There exists a point \( P \) on the right branch of the hyperbola such that \( \left( \overrightarrow{OP} + \overrightarrow{OF_{2}} \right) \cdot \overrightarrow{PF_{2}} = 0 \), where \( O \) is the origin. Additionally, \( \left| \overrightarrow{PF_{1}} \right| = \sqrt{3} \left| \overrightarrow{PF_{2}} \right| \). Determine the eccentricity of the hyperbola.

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