
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 $A B C A_{1} B_{1} C_{1}$ is inscribed in a sphere with base $A B C$ and lateral edges $A A_{1}$, $B B_{1}$, $C C_{1}$. Segment $C D$ 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