
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 $f(x) = \frac{1}{3}x^3 - ax^2 + bx$ ($a, b \in \mathbb{R}$), with $f'(0) = f'(2) = 1$.  
(1) Find the equation of the tangent line to the curve $y = f(x)$ at the point $(3, f(3))$;  
(2) If the function $g(x) = f(x) - 4x$, $x \in [-3, 2]$, find the intervals of monotonicity and the minimum value of $g(x)$.

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