
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]


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Given that the domain of the function $f(x)$ is $(-1, +\infty)$, and its derivative is $f'(x)$, the inequality $\frac{f(x)}{x+1}+\ln(x+1) \cdot f'(x) \geqslant \ln(x+1) \cdot f(x)$ always holds. Also, $f(4) = \frac{e^4}{\ln 5}$. Determine the solution set of the inequality $\ln(x+3) \cdot f(x+2) \geqslant e^{x-2}$.

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