
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 cylindrical barrel with radius $6$ feet and height $20$ feet is full of water. A solid cube with side length $12$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced, denoted as $v$, is required. Find $v^2$.

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