
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 $f(x) = \ln x - \frac{1}{2}ax^2 - bx$.

$(1)$ 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 $[1, e^2]$, find the range of the real number $m$.

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