
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 $a\in\mathbb{R}$.

(1) When $a=1$, find the equation of the tangent line to the graph of the function $f(x)$ at the point $(1,f(1))$.

(2) Does there exist a real number $a$ such that for any $x_{1}$, $x_{2}\in(0,+\infty)$ and $x_{1}\neq x_{2}$, the inequality $\frac{f(x_{2})-f(x_{1})}{x_{2}-x_{1}} > a$ always holds? If it exists, find the range of $a$; if not, explain the reason.

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