
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 $f(x)=\ln x-\frac{a}{{x+1}}$.  
$(1)$ Discuss the monotonicity of the function $f(x)$.  
$(2)$ If the function $f(x)$ has two extreme points $x_{1}$ and $x_{2}$, and $k{e^{f({{x_1}})+f({{x_2}})-4}}+\ln\frac{k}{{{x_1}+{x_2}-2}}≥0$ always holds, find the minimum value of the real number $k$.

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