
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
If the function $f(x)$ is a differentiable function defined on $(0,+\infty)$, with its derivative being $f'(x)$, and it satisfies $f'(x) + \frac{2}{x}f(x) > 0$, find the range of real number $a$ such that the inequality $\frac{{ax \cdot f(ax)}}{{\ln x}} \geq \frac{{f(\ln x) \cdot \ln x}}{{ax}}$ holds for all $x \in (1,+\infty)$.

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