
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 $l$ and $m$ are two different lines, and $\alpha$ and $\beta$ are two different planes. The following propositions are given:  
① If $l \subset \alpha$, $m \subset \alpha$, $l$ is parallel to $\beta$, and $m$ is parallel to $\beta$, then $\alpha$ is parallel to $\beta.  
② If $l \perp \alpha$ and $m \perp \alpha$, then $l$ is parallel to $m$.  
③ If $\alpha$ is parallel to $\beta$, $l \subset \alpha$, and $m \subset \beta$, then $l$ is parallel to $m.  
④ If $l \perp \alpha$, $m$ is parallel to $\beta$, and $\alpha \perp \beta$, then $l$ is perpendicular to $m.  
The correct proposition numbers are ____.

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