
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 $M$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$, points $A(a,0)$ and $B(0,-b)$, and the distance from the origin $O$ to line $AB$ is $\frac{2\sqrt{3}}{3}$.

(I) Find the equation of the ellipse $M$;

(II) Let line $l$: $y=2x+m$ intersect with ellipse $M$ at two distinct points $C$ and $D$. A line passing through point $E$ on segment $CD$ intersects with the $y$-axis at point $P$, such that $\overrightarrow{PE} \cdot \overrightarrow{CD}=0$ and $|\overrightarrow{PC}|=|\overrightarrow{PD}|$. Find the maximum value of the area $S$ of $\triangle PCD$.

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