
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
It is known that the line $l: y= \sqrt{3}x - 2\sqrt{3}$ passes through the right focus $F_2$ of the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), and the center of the ellipse $C$ is symmetrically opposite with respect to the line $l$ and lies on the line $x = \frac{a^2}{c}$ (where $2c$ is the focal distance). The line $m$ passes through the left focus $F_1$ of the ellipse $C$ and intersects the ellipse at points $M$ and $N$.

(1) Find the equation of the ellipse $C$;
(2) If $\overrightarrow{OM} \cdot \overrightarrow{ON} = \frac{2\lambda}{\tan \angle MON} \neq 0$ (where $O$ is the coordinate origin), find the maximum value of $\lambda$ when the line $m$ rotates around the point $F_1$.

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