
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 $f(x)=(ax+b)\ln x-bx+3$ has a tangent line at $(1,f(1))$ with the equation $y=2$.  
$(1)$ Find the values of $a$ and $b$;  
$(2)$ Find the extremum of the function $f(x)$;  
$(3)$ If $g(x)=f(x)+kx$ is monotonic at $(1,3)$, find the range of $k$.

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