
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 $f(x)= \frac {x}{|\ln x|}$.
$(1)$ Find the intervals of monotonicity for $f(x)$;
$(2)$ If the equation $f^{2}(x)-(2m+1)f(x)+m^{2}+m=0$ has $4$ distinct real roots, find the range of $m$;
$(3)$ If there exist positive real numbers $x$, $y$ with $x > y$, such that the inequality $ey(\sin \alpha+ \frac {1}{2}) > \frac {x}{|\ln x-\ln y|}$ holds, find the solution set for $\alpha$. (Here $e$ is the base of the natural logarithm)

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