
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 $f(x)=2\ln x-x^{2}$  
$(1)$ Discuss the monotonicity of $f(x)$ and find its maximum value;  
$(2)$ Let $g(x)=xe^{x}-(a-1)x^{2}-x-2\ln x$, if $f(x)+g(x)\geqslant 0$ always holds, find the range of the real number $a$.

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