
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 $$f(x)= \frac {1+2\ln x}{x^{2}}.$$  
(1) Find the intervals of monotonicity for the function $f(x)$.  
(2) Let $g(x)=ax^2-2\ln x-1$, if the function $y=g(x)$ has two distinct roots, find the range of the real number $a$.  
(3) If there exist $x_1, x_2 \in (0, +\infty)$ and $x_1 \neq x_2$, such that $$\frac {f(x_{1})-f(x_{2})}{\ln x_{1}-\ln x_{2}} \leq k$$ holds, find the range of the real number $k$.

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