
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
The following propositions are given:  
① The values of trigonometric functions with the same name for angles with the same terminal side are equal;  
② The values of trigonometric functions with the same name for angles with different terminal sides are unequal;  
③ If $\sin\alpha > 0$, then $\alpha$ is an angle in the first or second quadrant;  
④ If $\sin\alpha = \sin\beta$, then $\alpha = 2k\pi + \beta$, where $k \in \mathbb{Z}$;  
⑤ If $\alpha$ is an angle in the second quadrant, then $\frac{\alpha}{2}$ is an angle in the first quadrant. Among these propositions, the correct ones are numbered \_\_\_\_\_\_.

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