
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 $f(x)= \frac {x^{2}}{a}-2\ln x$ $(a\in \mathbb{R},a\neq 0)$.  
$(1)$ Discuss the monotonicity of the function $f(x)$;  
$(2)$ If the function $f(x)$ has a minimum value, denoted as $g(a)$, and the equation $g(a)+a- \frac {2}{9a}-1=m$ has three different real roots with respect to $a$, find the range of the real number $m$.

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