
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 the function $f(x) = x^3 + ax^2 + bx + c$, the tangent line of the curve $y = f(x)$ at the point $P(1, f(1))$ is $y = -4x + 1$, and the function $y = f(x)$ has an extremum at $x = 3$.

(1) Find the analytical expression of $f(x)$.

(2) Find the minimum value of $y = f(x)$ on the interval $[0, 4]$.

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