
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
A function $f(x) = \frac{1}{3}x^3 + cx + 3$ (where $c$ is a constant) is defined on $\mathbb{R}$. The tangent line to $f(x)$ at $x = 0$ is perpendicular to the line $y = x + 2$.

1. Find the analytical expression of the function $y = f(x)$.
2. Let $g(x) = 4\ln x - f'(x)$ (where $f'(x)$ is the derivative function of $f(x)$). Find the extreme value of $g(x)$.

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