
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 $f(x)=\frac{2x}{\ln x}$.
(1) Find the equation of the tangent line to the curve $y=f(x)$ that is perpendicular to the line $2x+y=0$;
(2) Find the interval(s) where $f(x)$ is strictly decreasing;
(3) If there exists $x\_0\in\[e,+∞)$, such that the function $g(x)=ae\ln x+\frac{1}{2}x^{2}-\frac{a+e}{2}\cdot\ln x\cdot f(x)\leqslant a$ holds true, find the range of values for the real number $a$.

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