
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
Let $M$ be any point on the ellipse $\frac{x^2}{4} + y^2 = 1$, $F\_1$ and $F\_2$ be the left and right foci of the ellipse, and $e$ be the eccentricity of the ellipse. The maximum value of $\sqrt{\frac{|MF\_1| \cdot |MF\_2|}{e^2}}$ is $\_\_\_\_\_\_.$

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