
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 $f(x)=(x-1)\ln x-(a+1)x$.
$(1)$ If $f(x)$ has an extremum at $x=1$, find $a$ and determine whether this extremum is a maximum or a minimum;
$(2)$ If $a=1$ and $f(x) > k$ always holds, find the maximum integer value of $k$.
Reference data: $\ln 2 \approx 0.69$, $\ln 3 \approx 1.10$, $\ln 3.6 \approx 1.28$

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