
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 $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\ (a > 0, b > 0)$, let $F\_1$ and $F\_2$ represent the foci, and $A\_1$ and $A\_2$ the vertices. A line parallel to one of the asymptotes of the hyperbola passes through focus $F\_2$ and intersects the other asymptote at point $M$. If point $M$ lies within the circle with $\overline{A\_1A\_2}$ as its diameter, determine the range of the hyperbola's eccentricity.

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