
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
A regular triangular prism \( ABC A_1 B_1 C_1 \) with a base \( ABC \) and side edges \( AA_1, BB_1, CC_1 \) is inscribed in a sphere of radius 3. The segment \( CD \) is the diameter of this sphere. Find the volume of the prism, given that \( AD = 2 \sqrt{6} \).

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