
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
$f(x)=2\ln x-ax-\frac{3}{ax}$. Discuss the monotonicity of $f(x)$; Let $g(x)=f(x)+{x}^{2}+\frac{3}{ax}$, if $g(x)$ has two extreme points $x_{1}$ and $x_{2}$, and $x_{1} \lt x_{2}$, find the maximum value of $g(x_{2})-2g(x_{1})$.

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