
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 right circular cone is sliced into three pieces by two planes, each parallel to the cone's base. The planes divide the cone such that the heights of the segments are in the ratio 1:1:2 from top to bottom. What is the ratio of the volume of the second-largest piece to the volume of the largest piece?

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