
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. Consider the following propositions: 
① If a line $l$ within plane $\alpha$ is perpendicular to any line within plane $\beta$, then $\alpha \perp \beta$;  
② If any line within plane $\alpha$ is parallel to plane $\beta$, then $\alpha \parallel \beta$;  
③ If plane $\alpha$ is perpendicular to plane $\beta$ and a line $l$ lies within plane $\alpha$, then $l \perp \beta$;  
④ If plane $\alpha$ is parallel to plane $\beta$ and a line $l$ lies within plane $\alpha$, then $l \parallel \beta$.  
The correct propositions are numbered __________.

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