
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 15 feet is full of water. A solid cube with side length 7 feet is set into the barrel so that one edge of the cube is vertical. Calculate the square of the volume of water displaced, $v^2$, when the cube is fully submerged.

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