
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
Let $\alpha$ and $\beta$ be two distinct planes, and let $m$ and $n$ be two distinct lines. Consider the following four propositions:  
① If $m \perpendicular \alpha$ and $n \subset \alpha$, then $m \perpendicular n$;  
② If $m \subset \alpha$, $n \subset \alpha$, $m \parallel \beta$, and $n \parallel \beta$, then $\alpha \parallel \beta$;  
③ If $\alpha \perpendicular \beta$, $\alpha \cap \beta = m$, $n \subset \alpha$, and $n \perpendicular m$, then $n \perpendicular \beta$;  
④ If $m \perpendicular \alpha$, $\alpha \perpendicular \beta$, and $m \parallel n$, then $n \parallel \beta$.  
Among these propositions, the correct ones are numbered as follows:

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