
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 $f(x)= \begin{cases} -\frac{1}{2}x+\frac{1}{4},x\in[0,\frac{1}{2}] \\ \frac{2x^{2}}{x+2},x\in(\frac{1}{2},1] \end{cases}, g(x)=a\sin(\frac{π}{3}x+\frac{3}{2}π)-2a+2 (a>0)$, provide the following conclusions:

(1) The range of the function $f(x)$ is $[0,\frac{2}{3}]$;

(2) The function $g(x)$ is increasing on $[0,1]$;

(3) For any $a>0$, the equation $f(x)=g(x)$ always has a solution in the interval $[0,1]$;

(4) 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 $\frac{4}{9}\leqslant a\leqslant \frac{4}{5}$.

The sequence numbers of all correct conclusions are \_\_\_\_\_\_.

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