
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 $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with two vertices $A(-a, 0)$, $B(a, 0)$, and a point $P$ on the ellipse distinct from $A$ and $B$. The slopes of the lines $PA$ and $PB$ are $k_{1}$ and $k_{2}$, respectively, with ${k_{1}} {k_{2}} = -\frac{1}{2}$.

(I) Find the eccentricity of the ellipse $C$.

(II) If $b = 1$, a line $l$ intersects the $x$-axis at point $D(-1, 0)$ and the ellipse at points $M$ and $N$. Find the maximum area of $\triangle OMN$.

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