
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 a hyperbola with the equation $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ and a point $C$ on it. A line passing through the center of the hyperbola intersects the hyperbola at points $A$ and $B$. The slopes of lines $AC$ and $BC$ are denoted as $k_{1}$ and $k_{2}$, respectively. Find the eccentricity of the hyperbola when $\frac{2}{k_{1}k_{2}}+\ln(k_{1}k_{2})$ is minimized.

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