
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
Point $P$ is located on the right branch of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, with its right focus $F_2$. $M$ is the midpoint of the line segment $PF_2$, and $|OF_2| = |F_2M|$. If $\overrightarrow{OF_2} \cdot \overrightarrow{F_2M} = \frac{1}{2}c^2$, find the eccentricity of the hyperbola.

A) $\frac{3}{2}$
B) $\frac{\sqrt{3} + 1}{2}$
C) $\sqrt{3}$
D) $\sqrt{5}$

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