
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 $F\_2$ be its left and right foci, respectively, and let $P$ be any point on the hyperbola. If the minimum value range of $\overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2}$ is $\left[- \frac{3}{4}c^{2}, - \frac{1}{2}c^{2}\right]$, find the range of the eccentricity of the hyperbola.

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