
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 $f(x) = 2f(-x) - x^2 - 12x - 1$ holds for any $x \in \mathbb{R}$,

(1) Find the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$; (2) Let $g(x) = f(x)e^{-x}$, find the intervals of monotonicity for $g(x)$.

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