
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 $f(x) = ax + x\ln x$, the slope of the tangent line at point $A(e, f(e))$ is $3$.  
(1) Find the value of $a$;  
(2) Determine the intervals on which $f(x)$ is monotonic;  
(3) If the inequality $f(x) - kx + k > 0$ holds for all $x \in (1, +\infty)$, find the largest integer value of $k$.

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