
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 $f(x)= \begin{cases} \frac {2x^{3}}{x+1}, & x\in \left( \frac {1}{2},1\right] \\ - \frac {1}{3}x+ \frac {1}{6}, & x\in [0, \frac {1}{2}] \end{cases}$, and the function $g(x)=a\sin \left( \frac {\pi}{6}x\right)-2a+2$ $(a > 0)$, find the range of the real number $a$ such that there exist $x_{1}$, $x_{2}\in[0,1]$ such that $f(x_{1})=g(x_{2})$.

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