
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 $y=x+ \frac {a}{x}$ ($a>0$) is monotonically decreasing in the interval $(0, \sqrt {a}]$ and monotonically increasing in the interval $[\sqrt {a}, +\infty)$; the function $h(x)=(x^{2}+ \frac {1}{x})^{3}+(x+ \frac {1}{x^{2}})^{3}$ ($x\in[ \frac {1}{2}, 2]$)  
(1) Please write down the monotonic intervals of the function $f(x)=x^{2}+ \frac {a}{x^{2}}$ ($a>0$) and the function $g(x)=x^{n}+ \frac {a}{x^{n}}$ ($a>0$, $n\in\mathbb{N}$, $n\geq3$) in $(0, +\infty)$ (only write the conclusion, no proof needed);  
(2) Find the maximum and minimum values of the function $h(x)$;  
(3) Discuss the number of real roots of the equation $h^{2}(x)-3mh(x)+2m^{2}=0$ ($0<m\leq30$).

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