
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 $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a>b>0)$ and a point $D(2, 1)$, if the line $OD$ intersects the ellipse $C$ at points $A$ and $B$, and $|AB|= \sqrt{2}|OD|$ ($O$ is the coordinate origin), and the eccentricity of the ellipse $C$ is $\frac{\sqrt{3}}{2}$.
(1) Find the standard equation of the ellipse $C$;
(2) If a line $l$ with slope $\frac{1}{2}$ intersects the ellipse $C$ at two distinct points $M$ and $N$, find the maximum area of $\triangle DMN$.

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