
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 $A(a,0)$, $B(0,-b)$, and the origin $O$ with a distance to line $AB$ of $\frac{2\sqrt{3}}{3}$.
(I) Find the equation of ellipse $M$;
(II) Let point $C$ be $(-a,0)$, and point $P$ be on ellipse $M$ (and distinct from points $A$ and $C$). Point $E$ is on line $PC$. If the equation of line $PA$ is $y = kx - 4$ and $\overrightarrow{CP} \cdot \overrightarrow{BE} = 0$, find the equation of line $BE$.

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