
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 $g(x)= \frac{x}{\ln x}$, $f(x)=g(x)-ax$

(Ⅰ) Find the monotonic intervals of the function $g(x)$;

(Ⅱ) If the function $f(x)$ is decreasing on $(1,+\infty)$, find the minimum value of the real number $a$;

(Ⅲ) If for every $x_1 \in [e,e^2]$, there exists $x_2 \in [e,e^2]$ such that $g(x_1) \leqslant f''(x_2)+2a$ holds, find the range of the real number $a$.

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