
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 $f(x)=\frac{3}{2}{x}^{2}-3\ln x$.  
$(1)$ Find the monotonic intervals of $f(x)$.  
$(2)$ Given that $m$ is an integer, the inequality $f(x\ln x+2x-1) \gt f(m(x-1))$ holds for $x \gt 1$ always. Find the maximum value of $m$.

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