
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 \( C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (\( a > 0, b > 0 \)), where \( l \) is an asymptote of the hyperbola \( C \), and \( F_{1}, F_{2} \) are the left and right focal points of the hyperbola \( C \). If the symmetric point of \( F_{1} \) about the line \( l \) lies on the circle \( (x-c)^{2}+y^{2}=c^{2} \) (where \( c \) is the semi-focal distance of the hyperbola), then the eccentricity of the hyperbola \( C \) is $\qquad$.

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