
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 that $m$ and $n$ are lines, and $\alpha$, $\beta$, $\gamma$ are planes, the following propositions are given:  
1. If $\alpha \perp \beta$, $\alpha \cap \beta = m$, $n \perp m$, then $n \perp \alpha$ or $n \perp \beta$;  
2. If $\alpha \parallel \beta$, $\alpha \cap \gamma = m$, $\beta \cap \gamma = n$, then $m \parallel n$;  
3. If $m$ is not perpendicular to $\alpha$, then $m$ cannot be perpendicular to countless lines within $\alpha$;  
4. If $\alpha \cap \beta = m$, $n \parallel m$; and $n \notin \alpha$, $n \notin \beta$, then $n \parallel \alpha$ and $n \parallel \beta$.  
Among these propositions, the correct ones are     . (Note: Fill in the numbers of the propositions you think are correct)

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