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

$(1)$ Find the analytic expression for $ f(x) $ and its interval of monotonic decrease.
$(2)$ Determine if there is a constant $ k $ such that for any $ x $ within the domain, $ f(x) > \frac {k}{\ln x}+2 \sqrt {x} $ always holds true. If such k exists, find its value; if not, explain why.

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