
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 the domain of the function $f(x)$ is $(0,+\infty)$, and its derivative is $f'(x)$, the inequality $(x+1)[2f(x)+xf'(x)] > xf(x)$ holds for all $x$. Also, $f(6) = \frac{7}{12}$. Find the solution set of the inequality $f(x+4) < \frac{3x+15}{{(x+4)}^{2}}$.

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