
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 its left and right foci denoted as $F_1$ and $F_2$ respectively. The points $M(-a, b)$, $N(a, b)$, $F_2$, and $F_1$ form an isosceles trapezoid with a height of $\sqrt{3}$ and an area of $3\sqrt{3}$.

(1) Find the equation of the ellipse.

(2) A line passing through point $F_1$ intersects the ellipse at points $A$ and $B$. Find the maximum area of $\triangle F_2AB$.

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