
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
(13 points) (2011•Chongqing) Let $f(x) = x^3 + ax^2 + bx + 1$, where the derivative $f'(x)$ satisfies $f'(1) = 2a$ and $f'(2) = -b$, with constants $a, b \in \mathbb{R}$.

(I) Find the equation of the tangent line to the curve $y = f(x)$ at the point $(1, f(1))$.

(II) Let $g(x) = f'(x)e^{-x}$. Find the extrema of the function $g(x)$.

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