
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 and an eccentricity of $\frac{2\sqrt{2}}{3}$, and one of its foci at $(0, 2\sqrt{2})$,
(1) Find the standard equation of the ellipse;
(2) Let a variable line $l$ passing through point $P(-1, 0)$ intersect with a circle $O$ centered at the origin with a radius of 2 at points $A$ and $B$. For a point $C$ on the ellipse and $\overrightarrow{AB} \cdot \overrightarrow{CP} = 0$, when $|\overrightarrow{CP}|$ reaches its maximum value, find the length of chord $AB$.

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