
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 $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$ and four points $P_1(4, 2)$, $P_2(2, 0)$, $P_3(-4, 3)$, and $P_4(4, 3)$, only three of these points lie on the hyperbola. Determine the eccentricity of the hyperbola. The options are:
A) $\frac{\sqrt{5}}{2}$
B) $\frac{5}{2}$
C) $\frac{\sqrt{7}}{2}$
D) $\frac{7}{2}$

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