
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 $f(x)=\ln x-mx^{2}+(1-2m)x+1$.  
$(1)$ If $m=1$, find the extreme value of $f(x)$;  
$(2)$ Discuss the monotonicity of $f(x)$;  
$(3)$ If $f(x) \leq 0$ holds for all $x \gt 0$, find the minimum value of the integer $m$.

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