
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
If $l, m, n$ are distinct lines in space, and $\alpha, \beta$ are non-coincident planes, then among the following propositions, the correct one is $\boxed{④}$.

$①.$ If $\alpha \parallel \beta, l \subset \alpha, n \subset \beta$, then $l \parallel n$  
$②.$ If $\alpha \perp \beta, l \subset \alpha$, then $l \perp \beta$  

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

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