
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 monotonically increasing function $f(x)$ is defined on the interval $(-1, +\infty)$. For any $x \in (-1, +\infty)$, the equation $f[f(x) - xe^x] = 0$ holds. Determine the interval in which the solution to the equation $f(x) - f'(x) = x$ lies.

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