
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
On the line $xy$, we mark the distances $A B' = b$ and $A C' = c$. At points $B'$ and $C'$, we draw perpendiculars to $xy$ and mark the segments $B' B = b'$ and $C' C = c'$. Calculate the volume of the solid of revolution formed by rotating the triangle $A B C$ around the line $xy$.

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