
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 $g(x)=ax$ ($a\in \mathbb{R}$).  
(I) If the graph of $y=f(x)$ and $y=g(x)=ax$ have no common points, find the range of the real number $a$.  
(II) Does there exist a real number $m$ such that for any $x\in\left(\frac{1}{2},+\infty\right)$, the graph of $y=f(x)+\frac{m}{x}$ is below the graph of $h(x)=\frac{e^x}{x}$? If it exists, find the maximum integer value of $m$; if not, explain why.  
(Reference data: $\ln 2=0.6931$, $\ln 3=1.0986$, $\sqrt{e}=1.6487$, $\sqrt[3]{e}=0.3956$)

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