
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 $f(x)=\ln x-2x$.  
$(1)$ Find the extreme values of the function $f(x)$;  
$(2)$ If $g(x)=\frac{1}{2}mx^{2}+(m-3)x-1(m∈R)$, does there exist an integer $m$ such that $f(x)\leqslant g(x)$ holds for any $x\in (0,+\infty)$? If yes, find the minimum value of $m$; if not, explain the reason.

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