
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 that the function $ f(x) = ax + x \ln x $ has a tangent slope of 3 at the point $ x = e $ (where $ e $ is the base of natural logarithms),  
(1) Calculate the value of the real number $ a $;  
(2) If $ k \in \mathbb{Z} $ and $ k < \frac{f(x)}{x-1} $ holds true for any $ x > 1 $, find the maximum value of $ k $.

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