
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
If a hyperbola $C$ is defined by $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\left(a \gt 0,b \gt 0\right)$ and one of its asymptotes is intersected by the circle $x^{2}+\left(y-2\right)^{2}=4$ to form a chord of length $2\sqrt{3}$, calculate the eccentricity of $C$.

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