
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 $m$, $n$, and $l$ are distinct lines, and $\alpha$, $\beta$, and $\gamma$ are distinct planes, consider the following propositions:  
1. If $\alpha \perp \beta$, $\alpha \cap \beta = m$, and $m \perp n$, then $n \perp \alpha$ or $n \perp \beta$  
2. If $\alpha \parallel \beta$, $\alpha \cap \gamma = m$, and $\beta \cap \gamma = n$, then $m \parallel n$  
3. If $m$ is not perpendicular to $\alpha$, then $m$ cannot be perpendicular to countless lines within $\alpha$  
4. If $\alpha \cap \beta = m$, $m \parallel n$, and $n \not\subset \alpha$, $n \not\subset \beta$, then $n \parallel \alpha$ and $n \parallel \beta$  
5. If $\alpha \cap \beta = m$, $\beta \cap \gamma = n$, $\alpha \cap \gamma = l$, and $\alpha \perp \beta$, $\alpha \perp \gamma$, $\beta \perp \gamma$, then $m \perp n$, $m \perp l$, $n \perp l$  
The correct propositions are numbered as follows:

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