
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 $f(x)=\frac{mx}{\ln x}$, the tangent line to the curve $y=f(x)$ at the point $\left( {{e}^{2}},f({{e}^{2}}) \right)$ is perpendicular to the line $2x+y=0$ (where $e$ is the base of the natural logarithm).

(I) Find the analytical expression of $f(x)$ and its monotonically decreasing interval;

(II) Determine if there exists a constant $k$ such that for any $x$ in the domain of the function, $f(x) > \frac{k}{\ln x}+2\sqrt{x}$ always holds. If such a constant exists, find its value; otherwise, explain why.

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