
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
In  $ xy$  plane, find the minimum volume of the solid by rotating the region boubded by the parabola  $ y \equal{} x^2 \plus{} ax \plus{} b$  passing through the point  $ (1,\ \minus{} 1)$  and the  $ x$  axis about the  $ x$  axis

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