
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 $\frac{x^{2}}{8}+\frac{y^{2}}{2}=1$, let $C$ be the right vertex of the ellipse, $A$ be any point on the ellipse in the first quadrant, and $B$ be the point symmetric to $A$ with respect to the origin. A perpendicular from point $A$ to the $x$-axis intersects line $BC$ at point $D$. Compare the sizes of $|AC|^{2}$ and $|CD| \cdot |CB|$, and provide a proof.

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