
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 $C:\frac{y^2}{a^2}-\frac{x^2}{b^2}=1(a>0,b>0)$, where $F$ is the lower focus of $C$. $O$ is the origin, $l_{1}$ is the asymptote of $C$ with a slope greater than 0. A line passing through $F$ with a slope of $\frac{\sqrt{3}}{3}$ intersects $l_{1}$ at point $A$ and the positive $x$-axis at point $B$. If $|OA|=|OB|$, calculate the eccentricity of $C$.

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