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

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