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

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