
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 $f\left( x \right)=a\ln x+\frac{1}{x}$, where $a$ is a real constant.

(1) Discuss the extreme values of the function $f\left( x \right)$;

(2) When $x=1$ is the extreme point of the function $f\left( x \right)$, let $g\left( x \right)=f\left( x \right)-\frac{1}{x}$, compare the size of $\frac{g\left( n \right)-g\left( m \right)}{2}$ and $\frac{n-m}{n+m}$ for $0 < m < n$, and explain the reason.

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