
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 $y=x+ \frac {t}{x}$ has the following properties: If the constant $t>0$, then the function is decreasing on $(0, \sqrt {t}]$ and increasing on $[\sqrt {t}, +\infty)$.  
(1) If $f(x)=x+ \frac {a}{x}$, and the function has a minimum value of 4 on $(0, a]$, find the value of $a$;  
(2) For the function in (1), if its range on interval $A$ is $[4, 5]$, find the interval $A$ with the maximum length (Note: interval length = right endpoint of the interval - left endpoint of the interval);  
(3) If the domain of the function in (1) is $[2, +\infty)$, solve the inequality $f(a^2-a) \geq f(2a+4)$.

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