
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
Consider an ellipse (C) with the equation \\(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)\\) and an eccentricity of \\(\frac{\sqrt{3}}{3}\\). A line (l) passing through the left focus intersects the upper half of the ellipse at point M. When the slope of line (l) is \\(\frac{\sqrt{3}}{3}\\), the distance |FM| equals \\(\frac{4\sqrt{3}}{3}\\).

1. Find the equation of the ellipse (C).
2. Points A and B on the ellipse (C) are symmetric about the line (l). Find the maximum area of triangle AOB.

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