
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 a focal length of $2\sqrt{3}$, line $l_1$: $y = kx (k \neq 0)$ intersects the ellipse at points $A$ and $B$. Line $l_2$, passing through point $B$ with a slope of $\frac{1}{4}k$, intersects the ellipse $C$ at another point $D$, and $AD \perp AB$.
(1) Find the equation of ellipse $C$;
(2) Let line $l_2$ intersect the $x$-axis and $y$-axis at points $M$ and $N$, respectively. Find the maximum area of $\triangle OMN$.

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