
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 the function $f\left(x\right)=\frac{1}{1+{x}^{2}}-\ln |x|$, if the inequality $f\left(-ax+\ln x+1\right)+f\left(ax-\ln x-1\right)\geqslant 2f\left(1\right)$ holds for $x\in \left[1,3\right]$, then the range of real number $a$ is ______.

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