
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
Consider a cylindrical tank with radius $3$ feet and height $12$ feet. Half of a solid cube with side length $6$ feet is submerged vertically in the tank so that the space diagonal of the cube remains vertical. Compute the volume of the water displaced by the submerged portion of the cube, denoted as $v$. Then find $v^3$.

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