
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 the 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 passing through the right focus (F2(c,0)) perpendicular to the x-axis intersects the ellipse at points A and B, such that |AB|=\\(\frac{4\sqrt{3}}{3}\\). Additionally, a line (l) passing through the left focus (F1(-c,0)) intersects the ellipse at point M.

1. Find the equation of the ellipse (C).
2. Given that points A and B on the ellipse (C) are symmetric with respect to line (l), find the maximum area of triangle AOB.

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