
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
Consider a hyperbola $C$: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$. Let $M$ be a point on the hyperbola $C$ and the center of a circle. This circle is tangent to the $x$-axis at one focus $F$ of the hyperbola and intersects the $y$-axis at points $P$ and $Q$. If $\triangle MPQ$ is an equilateral triangle, find the eccentricity of the hyperbola $C$.

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