
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
The eccentricity of the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \left(a > b > 0\right)$ is $\dfrac{\sqrt{3}}{3}$. A line perpendicular to the $x$-axis passing through the right focus $F_2(c,0)$ intersects the ellipse at points $A$ and $B$, and $|AB| = \dfrac{4\sqrt{3}}{3}$. Another line $l$ passing through the left focus $F_1(-c,0)$ intersects the ellipse at point $M$.

(Ⅰ) Find the equation of ellipse $C$;

(Ⅱ) Points $A$ and $B$ on ellipse $C$ are symmetric about line $l$. Find the maximum value of the area of $\triangle AOB$.

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