
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 a function $f(x) = \ln x$ and $g(x) = \frac{m(x+n)}{x+1}$ where $m>0$,  
(1) When $m=1$, the tangent lines of the functions $y=f(x)$ and $y=g(x)$ at $x=1$ are perpendicular. Find the value of $n$.  
(2) If the function $y=f(x)-g(x)$ is not monotonic in its domain, determine the range of values for $m-n$.  
(3) Does there exist a real number $a$ such that $f\left(\frac{2a}{x}\right) \cdot f\left(e^{ax}\right) + f\left(\frac{x}{2a}\right) \leq 0$ holds true for all positive real numbers $x$? If such an $a$ exists, find it; if not, explain why.

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