
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 $f(x)=x^{2}-\frac{a}{2}\ln x$, the slope of the tangent line at the point $\left(\frac{1}{2},f\left( \frac{1}{2} \right)\right)$ is $0$.

$(1)$ Discuss the monotonicity of the function $f(x)$;

$(2)$ If $g(x)=f(x)+\frac{1}{2}mx$ has no zeros in the interval $(1,+\infty)$, find the range of the real number $m$.

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