
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 an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with left and right foci $M$ and $N$. If a pair of hyperbolas is drawn with the foci of the ellipse as its vertices and the vertices of the ellipse's major axis as its foci, it forms an equilateral hyperbola.

(1) Find the eccentricity of the ellipse.

(2) Let $L$ be a line passing through the right focus $N$ of the ellipse, intersecting the ellipse at points $P$ and $Q$. When the perimeter of $\triangle MPQ$ is $8$, find the maximum area of $\triangle MPQ$.

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