
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 with its foci on the $y$-axis, an eccentricity of $\frac{2\sqrt{2}}{3}$, and one focus at $(0, 2\sqrt{2})$.
(1) Find the standard equation of the ellipse;
(2) A moving line $l$ passes through point $P(-1,0)$, intersecting a circle $O$ centered at the origin with a radius of $2$ at points $A$ and $B$. $C$ is a point on the ellipse such that $\overrightarrow{AB} \cdot \overrightarrow{CP} = 0$. Find the length of chord $AB$ when $|\overrightarrow{CP}|$ is at its maximum.

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