
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
Let $f(x)=\frac{1-a}{2}x^{2}+ax-\ln x$ $(a\in \mathbb{R})$.<br/>$(1)$ When $a \gt 1$, discuss the monotonicity of the function $f(x)$;<br/>$(2)$ If for any $a\in (3,4)$ and any $x_{1}$, $x_{2}\in [1,2]$, the inequality $\frac{({a}^{2}-1)m}{2}+\ln 2 \gt |f(x_{1})-f(x_{2})|$ holds, find the range of real number $m$.

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