
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 an isosceles triangle \(ABC\), the base \(BC\) is extended beyond \(C\) to a point \(D\) such that \(BC = CD\). A perpendicular is drawn at \(D\) to the line \(BD\), and the plane is rotated around this perpendicular. Determine the surface area and volume of the resulting solid of revolution described by the triangle \(ABC\), given that the length of side \(AB\) is \(b\) and the angle \(\angle BAC = \alpha\).

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