
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 $f(x) = \frac{1}{2}x^{2} + a\ln x$, if for any two distinct positive real numbers $x_{1}$, $x_{2}$, it always holds that $\frac{f(x_{1})-f(x_{2})}{x_{1}-x_{2}} > 2$, then the range of the real number $a$ is ______.

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