
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 the ellipse \(\frac{x^{2}}{8}+\frac{y^{2}}{2}=1\), with \(C\) as the right vertex, let \(A\) be any point on the ellipse in the first quadrant, and \(B\) be the point symmetric to \(A\) about the origin. A perpendicular line is drawn from \(A\) to the \(x\)-axis, intersecting \(BC\) at point \(D\). Compare the magnitudes of \(|AC|^{2}\) and \(|CD||CB|\), providing a proof.

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