
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 cone has a base with a circumference of \(24\pi\) inches and a height of 40 inches. The height of this cone is reduced while the circumference stays the same. The volume of the shorter cone is \(432\pi\) cubic inches. What is the ratio of the shorter height to the original height? Express your answer as a common fraction.

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