
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
Determine the side length of a regular dodecagon inscribed in a circle.

Al-Karaji provides a verbal expression, according to which:

$$
a_{12}^{2}=\left[\frac{d}{2}-\sqrt{\left(\frac{d}{2}\right)^{2}-\left(\frac{d}{4}\right)^{2}}\right]^{2}+\left(\frac{d}{4}\right)^{2}
$$

Verify the correctness of this expression.

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