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

(I) When $a=\frac{1}{4}$, find the maximum value of $f(x)$;

(II) Let $g(x)=f(x)+ax^{2}+\frac{1}{2}x+\frac{a}{x}$, where $x\in(0,3]$, the slope $k$ of the tangent line at any point $P(x_{0},y_{0})$ on its graph satisfies $k\leqslant\frac{1}{2}$. Find the range of values for the real number $a$;

(III) When $a=0$, the equation $2mf(x)=x(x-3m)$ has a unique real solution. Find the value of the positive real number $m$.

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