
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 an ellipse (C) given by the equation $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$. A line $(l)$ intersects the ellipse at points $P$ and $Q$, with $M$ as the midpoint of segment $PQ$ and $O$ as the origin of the coordinate system. Let the slope of line $(l)$ be $k_{1}$, and the slope of line $(OM)$ be $k_{2}$, where $k_{1}k_{2} = -\frac{2}{3}$.

1. Find the eccentricity of ellipse $(C)$.
2. Given that line $(l)$ intersects the $x$-axis at point $D(-\sqrt{3}, 0)$ and satisfies $\overrightarrow{DP} = 2 \overrightarrow{QD}$, find the equation of ellipse $(C)$ when the area of triangle $OPQ$ is maximum.

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