
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 $20\pi$ inches and an initial height of 50 inches. The height of this cone is reduced while keeping the circumference unchanged. The volume of the shortened cone is $500\pi$ cubic inches. What is the ratio of the shortened 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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