
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 $f(x)$ is a linear function, and $f(x) = x \int_{0}^{2} f(t) \, dt + 1$,  
(1) Find the expression for the function $f(x)$;  
(2) If $g(x) = x \cdot f(x)$, find the volume of the solid of revolution generated by revolving the area enclosed by the curve $y = g(x)$ and the x-axis around the x-axis.

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