
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 the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{b^{2}} = 1 (b \in \mathbb{N}^*)$, with foci $F\_1$ and $F\_2$, and a point $P$ on the hyperbola where $|OP| < 5$. If $|PF\_1|$, $|F\_1F\_2|$, and $|PF\_2|$ form a geometric sequence, what is the eccentricity of the hyperbola?

A) $2$
B) $3$
C) $\frac{5}{3}$
D) $\frac{\sqrt{5}}{2}$

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