
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 $f(x)=\frac{1}{2}\ln x-x$.  
$(1)$ Find the equation of the tangent line of $f(x)$ at $x=1$.  
$(2)$ If $g(x)=f(x)+mx^{2}$, where $x_{1}$ and $x_{2}$ are the two extreme points of the function $g(x)$, find the range of $g(x_{1})+g(x_{2})$.

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