
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 a line $l$ perpendicular to plane $\alpha$, and a line $m$ contained in plane $\beta$, consider the following propositions:

$①$ If $\alpha \parallel \beta$, then $l \perp m$;  
$②$ If $\alpha \perp \beta$, then $l \parallel m$;

$③$ If $l \parallel m$, then $\alpha \perp \beta$;  
$④$ If $l \perp m$, then $\alpha \parallel \beta$.

Among these propositions, the correct ones are \_\_\_\_\_\_\_.

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