
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 $\alpha$ and $\beta$ are two non-intersecting planes, and $m$ and $n$ are two distinct lines. Select the correct proposition(s) from the following options:

$①$ If $\alpha \parallel \beta$ and $m \subset \alpha$, then $m \parallel \beta$;

$②$ If $m \parallel \alpha$ and $n \subset \alpha$, then $m \parallel n$;

$③$ If $\alpha \perp \beta$, $\alpha \bigcap \beta = n$, and $m \perp n$, then $m \perp \beta$;

$④$ If $n \perp \alpha$, $n \perp \beta$, and $m \perp \alpha$, then $m \perp \beta$.

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