
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
In the Cartesian coordinate plane $(xOy)$, let $F_1$ and $F_2$ be the left and right foci, respectively, of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. Suppose there exists a point $P$ on the right branch of the hyperbola such that $\overrightarrow{OP} \cdot \overrightarrow{F_2P}=\overrightarrow{F_2O} \cdot \overrightarrow{F_2P}$, and $|\overrightarrow{PF_1}|=\sqrt{3}|\overrightarrow{PF_2}|$. Find the eccentricity of this hyperbola.

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