
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
Let the function $f(x) = \ln x$, and $g(x) = \frac{m(x+n)}{x+1}$ ($m>0$).  
(1) When $m=1$, the tangent lines of $y=f(x)$ and $y=g(x)$ at $x=1$ are perpendicular to each other, find the value of $n$;  
(2) If for any $x>0$, $|f(x)| \geq |g(x)|$ always holds, find the value of the real number $n$ and the maximum value of the real number $m$.

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