
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 $f(x)= \sqrt {x}|x-a|$, where $a\in\mathbb{R}$, and $g(x)=16x^{3}+mx^{2}-15x-2$, with $g(2)=0$.
- (I) Find the extreme values of the function $g(x)$;
- (II) If the function $f(x)$ is a monotonic function, find the range of the real number $a$;
- (III) Let $a > 0$, if there exists a real number $t(t > a)$, such that when $x\in[0,t]$, the range of the function $f(x)$ is $[0, \frac {t}{2}]$, find the range of the real number $a$.

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