
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 $f(x)= \frac {1}{2}f'(1)x+x\ln x$  
$(1)$ Find the extreme values of the function $f(x)$;  
$(2)$ If $k\in\mathbb{Z}$, and $f(x) > k(x-1)$ holds for any $x\in(1,+\infty)$, find the maximum value of the integer $k$.

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