
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 maximum value of the function $f(x)$;

$(2)$ Let $F(x)=f(x)+\frac{1}{2} x^{2}+bx+\frac{a}{x} (0 < x\leqslant 3)$. If the slope $k$ of the tangent line at any point $P(x_{0},y_{0})$ on its graph is always less than or equal to $\frac{1}{2}$, find the range of the real number $a$;

$(3)$ When $a=0$, $b=-1$, the equation $x^{2}=2mf(x)$ (where $m > 0$) has a unique real solution, find the value of $m$.

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