
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 triangle is rotated about one leg generating a volume of \( 500\pi \; \text{cm}^3 \) for the resulting cone. When rotated about the other leg, the volume is \( 1620\pi \; \text{cm}^3 \). Compute the length (in cm) of the hypotenuse of this triangle.

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