
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 \(ABC A_1 B_1 C_1\) has been inscribed in a sphere, with \(ABC\) as its base and lateral edges \(A A_1, B B_1, C C_1\). The segment \(CD\) is the diameter of this sphere. Point \(K\) is the midpoint of edge \(A A_1\), and point \(L\) is the midpoint of edge \(A B\). Find the volume of the prism if \(D L = \sqrt{6}\) and \(D K = 3\).

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