
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 $f(x) = \frac{1}{2}x^2 - 2a\ln x + (a-2)x$, where $a \in \mathbb{R}$.
(i) When $a = -1$, determine the extreme values of the function $f(x)$.
(ii) When $a < 0$, discuss the monotonicity of the function $f(x)$.
(iii) Does there exist a real number $a$ such that for any $m, n \in (0, +\infty)$ and $m \neq n$, the inequality $\frac{f(m)-f(n)}{m-n} > a$ is always valid? If such $a$ exists, find the range of $a$; if not, explain why.

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