
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 $f(x)=\frac{{alnx}}{{x+1}}+\frac{b}{x}$, the tangent line equation of the curve $y=f\left(x\right)$ at the point $\left(1,f\left(1\right)\right)$ is $x+2y-3=0$.  
$(1)$ Find the values of $a$ and $b$;  
$(2)$ If $f(x)>\frac{{lnx}}{{x-1}}+\frac{k}{x}$ when $x>1$, find the range of values for $k$.

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