
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 $f(x) = \frac{1}{2}x^2 - ax + a\ln x$ $(a \neq 0)$.
(I) Find the monotonic intervals of the function $f(x)$;
(II) If $a \geq \frac{9}{2}$, and $x_1$, $x_2$ ($x_1 < x_2$) are the two extreme points of the function $f(x)$, find the minimum value of $f(x_1) - f(x_2)$.

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