
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 $f(x)=\ln \frac{x+1}{x-1}$.

$(1)$ Determine the parity of the function $f(x)$ and provide a proof;  
$(2)$ If the function $g(x)=\ln x-(x-1)$ is monotonically decreasing on $(1,+\infty)$, compare the size relationship between $f(2)+f(4)+\ldots+f(2n)$ and $2n$ $(n\in\mathbb{N}^{*})$, and explain the reason.

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