
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
(1) Given that the derivative of the function $f(x)$ is $f'(x)$, and it satisfies $f(x)=2xf'(e)+\ln x$, find $f'(e)=$ ______.

(2) The solution set of the inequality $|x+2|\leqslant 5$ is ______.

(3) Let $x > 0$, the minimum value of $x^2+\frac{2}{x}$ is ______.

(4) The equation of the tangent line to the curve $y=e^x$ at the point $(1,e)$ is ______. (Note: $e$ is the base of the natural logarithm)

(5) If the function $f(x)=x^2+2x+a\ln x$ is monotonically decreasing on the interval $(0,1)$, then the range of the real number $a$ is ______.

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