
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 functions $f(x)=\ln x$ and $g(x)=\frac{m(x+n)}{x+1} (m > 0)$.

$(1)$ If the function $y=f(x)$ and $y=g(x)$ have the same tangent line at $x=1$, find the value of $m$;

$(2)$ If for all $x \geqslant 1$, the inequality $|f(x)| \geqslant |g(x)|$ always holds, find the maximum value of the real number $m$.

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