
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 $f(x)= \frac {a\ln x+b}{x}$ where $a\leqslant 2$ and $a\neq 0$, and $f(x)$ has an extremum at $x= \frac {1}{e}$.  
$(1)$ Find the intervals of monotonicity for the function $f(x)$.  
$(2)$ If the graph of the function $f(x)$ and the graph of the function $g(x)=a+2-x- \frac {2}{x}$ have exactly one intersection point in the interval $(0,2]$, find the range of the real number $a$.

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