
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 $m,n$ are lines, and $\alpha, \beta, \gamma$ are planes, consider the following statements; the correct statement numbers are __________

$①$ If $\alpha \perp \beta$, $\alpha \bigcap \beta = m$, $n \perp m$, then $n \perp \alpha$ or $n \perp \beta$;  
$②$ If $\alpha \parallel \beta$, $\alpha \bigcap \gamma = m$, $\beta \bigcap \gamma = n$, then $m \parallel n$;  
$③$ If $m$ is not perpendicular to $\alpha$, then $m$ cannot be perpendicular to countless lines within $\alpha$;  

$④$ If $\alpha \bigcap \beta = m$, $n \parallel m$ and $n \not\subset \alpha$, $n \not\subset \beta$, then $n \parallel \alpha$ and $n \parallel \beta$．

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