
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 $f(x)=ax+x\ln x$ where its graph has a tangent line with a slope of $3$ at the point $x=e$ ($e$ is the base of the natural logarithm),

$(1)$ Find 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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