
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: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) with eccentricity $e = \sqrt{5}$, and $A_1$ and $A_2$ are the left and right vertices of the hyperbola C respectively, find the value of $b$ if the area of the quadrilateral $MA_2NA_1$ is $4$.

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