
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 a radius of 5 feet and a height of 10 feet is full of water. A regular tetrahedron with edge length 10 feet is placed into the barrel so that its vertex is at the center of the barrel's bottom and its axis is vertical and halfway submerged. Calculate the square of the volume $v^2$ of water displaced by the tetrahedron.

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