
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 that $f'(x)$ is the derivative of the function $f(x)$ ($x \in \mathbb{R}$), satisfying $f'(x) = f(x)$, and $f(0) = 2$. Let the function $g(x) = f(x) - \ln f^3(x)$ have a zero at $x_0$, determine the interval in which $x_0$ lies.

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