
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 $f(x)=\frac{-4x+5}{x+1}$ and $g(x)=a\sin(\frac{\pi}{3}x)-2a+2(a>0)$, if there exist $x_{1}$ and $x_{2}\in \left[0,2\right]$ such that $f(x_{1})=g(x_{2})$, then the range of real number $a$ is ______

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