
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 $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0, b>0)$, and the point $F_{1}$ is the symmetric point of $F_{2}$ with respect to the asymptote and lies exactly on a circle with center at $F_{2}$ and radius $|OF_{2}|$, calculate the eccentricity of the hyperbola.

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