
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 being $F_{1}$ and $F_{2}$ respectively, and four 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 two points $A$ and $B$. Find the maximum area of $\triangle F_{2}AB$.

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