
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 $g(x)= \frac {x}{\ln x}$, $f(x)=g(x)-ax$.  
$(1)$ Find the intervals of monotonicity for the function $g(x)$.  
$(2)$ If the function $f(x)$ is decreasing on $(1,+\infty)$, find the minimum value of the real number $a$.  
$(3)$ If there exist $x_{1}$, $x_{2}\in[e,e^{2}]$ such that $f(x_{1})\leqslant f′(x_{2})+a$, find the range of the real number $a$.

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