
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
Given  $ \triangle{ABC}$ , where  $ A$  is at  $ (0,0)$ ,  $ B$  is at  $ (20,0)$ , and  $ C$  is on the positive  $ y$ -axis. Cone  $ M$  is formed when  $ \triangle{ABC}$  is rotated about the  $ x$ -axis, and cone  $ N$  is formed when  $ \triangle{ABC}$  is rotated about the  $ y$ -axis. If the volume of cone  $ M$  minus the volume of cone  $ N$  is  $ 140\pi$ , find the length of  $ \overline{BC}$ .

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