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

A) $(\frac{\sqrt{6} + \sqrt{2}}{2}, +\infty)$
B) $(\frac{\sqrt{5} + 1}{2}, \frac{\sqrt{6} + \sqrt{2}}{2})$
C) $(\sqrt{6} + \sqrt{2}, +\infty)$
D) $(1, \sqrt{6} + \sqrt{2})$

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