
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
Let ellipse M be defined by the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\frac{\sqrt{2}}{2}$. Let point A be $(a, 0)$, point B be $(0, -b)$, and the distance from the origin O to the line AB be $\frac{2\sqrt{3}}{3}$.

(I) Find the equation of ellipse M.

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

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