
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 $f(x)=\ln x-\frac{1}{2}a{x^{2}}+x$ $(a\in\mathbb{R})$.

$(1)$ When $a=2$, find the maximum value of the function $f(x)$;

$(2)$ Let $F(x)=f(x)+\frac{1}{2}ax^{2}-x+\frac{a}{x}$ $(0 < x\leqslant 3)$, if the slope $k$ of the tangent line at any point $P(x_{0},y_{0})$ on its graph satisfies $k\leqslant \frac{1}{2}$ always holds, find the range of the real number $a$;

$(3)$ When $a=0$, the equation $mf(x)=x^{2}$ has a unique real solution, find the value of the positive number $m$.

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