
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
A circle with its center at point M on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0$, $b > 0$) is tangent to the x-axis exactly at one of the hyperbola's foci F, and intersects the y-axis at points P and Q. If $\triangle MPQ$ is an equilateral triangle, find the eccentricity of the hyperbola.

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