
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 $f(x)=a\sin \frac{\pi }{6}x-2a+3$ ($a > 0$), and the function $g(x)=\frac{2x}{{x}^{2}+x+2}$, $x\in[0,1]$, if there exist $x_{1}$, $x_{2}\in[0,1]$ such that $f(x_{1})=g(x_{2})$ holds, then the range of the real number $a$ is \_\_\_\_\_\_\_\_.

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