
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]


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Given functions $f(x)=\left\{\begin{array}{l}\frac{x}{x+2},x∈(\frac{1}{2},1]\\-\frac{1}{2}x+\frac{1}{4},x∈[0,\frac{1}{2}]\end{array}\right.$ and $g(x)=asin(\frac{π}{3}x+\frac{3π}{2})-2a+2(a＞0)$, provide the following conclusions:<br/>① The range of function $f\left(x\right)$ is $[0,\frac{1}{3}]$;<br/>② The function $g\left(x\right)$ is increasing on $\left[0,1\right]$;<br/>③ For any $a \gt 0$, the equation $f\left(x\right)=g\left(x\right)$ always has a solution in $\left[0,1\right]$;<br/>④ If there exist $x_{1}$ and $x_{2}\in \left[0,1\right]$, such that $f(x_{1})=g(x_{2})$ holds, then the range of real number $a$ is $\frac{5}{9}≤a≤\frac{4}{5}$.<br/>The correct conclusions are represented by the following numbers: ____.

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