
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 $f(x)=\ln x+ \frac{m}{2x}$ and $g(x)=x-2m$, where $m\in R$ and $e=71828…$ is the base of the natural logarithm.

(1) When $m=1$, find the minimum value of the function $f(x)$.

(2) For all $x\in[\frac{1}{e},1]$, does there exist $m\in(\frac{1}{2},1)$ such that $f(x) > g(x)+1$ holds? If so, find the range of $m$; if not, explain the reason.

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