
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
Let the function $f(x)=\ln x-\frac{1}{2}a{x}^{2}-bx$.

$(1)$ When $a=b=\frac{1}{2}$, find the intervals of monotonicity for the function $f(x)$;

$(2)$ When $a=0$, $b=-1$, the equation $f(x)=mx$ has a unique real solution in the interval $\left[ 1,{e}^{2} \right]$. Find the range of the real number $m$.

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