
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
The hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0, b > 0$) has its left and right foci denoted by $F_1$ and $F_2$, respectively. Suppose there exists a point $P$ on $C$ such that $|PO| = \frac{1}{2}|F_1F_2|$, where $O$ is the origin, and the slope of line $OP$ is $\sqrt{3}$. Then, the eccentricity of the hyperbola $C$ is \_\_\_\_\_.

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