
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
Consider a circle with center $O$ and a chord $AB$ of length $\sqrt{3}$, where the diameter of the circle is 2. Let $C$ be a point on the circle such that $\angle AOB = 120^\circ$. If point $D$ lies on the circle and segment $\overline{CD}$ is perpendicular to $\overline{AB}$, find the length of segment $CD$.

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