
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 that $F_1$ and $F_2$ are the left and right foci of the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, there exists a point $P$ on the right branch of the hyperbola such that in $\triangle F_2PF_1$, $|F_2P| = |F_2F_1|$. The angle bisector of $\angle F_2PF_1$ intersects line segment $F_1F_2$ at point $Q$ with $\frac{F_1Q}{QF_2} = \frac{3}{2}$. Determine the eccentricity of the hyperbola.

A) $2$
B) $3$
C) $4$
D) $5$

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