
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 $f(x)=({2a-1})\ln x-\frac{1}{x}-2ax(a\in R)$.<br/>$(1)$ When $a=0$, find the monotonicity of the function $f(x)$;<br/>$(2)$ When $a\neq 0$, discuss the monotonicity of the function $f(x)$;<br/>$(3)$ If for any $a\in \left[-2,-1\right)$, when $x_{1}$, $x_{2}\in \left[1,e\right]$, the inequality $({m-2e})a-\frac{1}{e}+2≥|{f({{x_1}})-f({{x_2}})}|$ always holds, find the range of real number $m$.

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