
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 two lines $m$ and $n$, and two planes $\alpha$ and $\beta$, consider the following four propositions:  
① If $m \parallel n$ and $m \perp \alpha$, then $n \perp \alpha$;  
② If $\alpha \parallel \beta$, $m \subset \alpha$, and $n \subset \beta$, then $m \parallel n$;  
③ If $m \parallel n$ and $m \parallel \alpha$, then $n \parallel \alpha$;  
④ If $\alpha \parallel \beta$, $m \parallel n$, and $m \perp \alpha$, then $n \perp \beta$.  
Among these propositions, the correct ones are.

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