
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
Among the following four propositions, the correct ones are \_\_\_\_\_\_ (fill in the serial numbers of all correct propositions)  
① If real numbers $a$, $b$, $c$ satisfy $a+b+c=3$, then at least one of $a$, $b$, $c$ is not less than 1  
② If $z$ is a complex number, and $|z|=1$, then the maximum value of $|z-i|$ is 2  
③ For any $x\in(0, +\infty)$, it holds that $x>\sin x$  
④ The definite integral $\int_{0}^{\sqrt{\pi}} \sqrt{\pi-x^{2}}dx= \frac{\pi^{2}}{4}$.

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