
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
If the function $f(x)$ is an increasing function on some interval $I$ within its domain $D$, and $F(x)= \frac {f(x)}{x}$ is a decreasing function on $I$, then $y=f(x)$ is called a "non-perfect increasing function" on $I$. Given $f(x)=\ln x$, $g(x)=2x+ \frac {2}{x}+a\ln x(a\in\mathbb{R})$  
$(1)$ Determine whether $f(x)$ is a "non-perfect increasing function" on $(0,1]$;  
$(2)$ If $g(x)$ is a "non-perfect increasing function" on $[1,+\infty)$, find the range of the real number $a$.

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