
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 line $l$, line $m$, plane $\alpha$, and plane $\beta$, with $l \perpendicular \alpha$, $m \subset \beta$, the following four propositions are given:  
① If $\alpha \parallel \beta$ then $l \perpendicular m$;  
② If $l \perpendicular m$ then $\alpha \parallel \beta$;  
③ If $\alpha \perpendicular \beta$, then $l \parallel m$;  
④ If $l \parallel m$ then $\alpha \perpendicular \beta$. Among these, the correct propositions are.

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