
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 $f(x) = a\ln x + \frac{1}{2}x^2$ ($a > 0$), if for any two different positive real numbers $x_1$, $x_2$, it always holds that $\frac{f(x_1) - f(x_2)}{x_1 - x_2} \geq 2$, determine the range of the parameter $a$.

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