
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 $f(x) = x \ln x + ax$ ($a \in \mathbb{R}$),  
(I) If $f(x)$ is an increasing function on the interval $[e^2, +\infty)$, find the range of $a$;  
(II) If for any $x \in (1, +\infty)$, $f(x) > k(x-1) + ax - x$ always holds, find the value of the positive integer $k$.

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