
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 different lines, and $\alpha$ and $\beta$ are different planes, then the correct proposition number(s) is/are.

$①$ If $m\parallel n$ and $m\perp\beta$, then $n\perp\beta$;  
$②$ If $m\parallel n$ and $m\parallel\beta$, then $n\parallel\beta$;  
$③$ If $m\parallel\alpha$ and $m\parallel\beta$, then $\alpha\parallel\beta$;  
$④$ If $n\perp\beta$ and $n\perp\alpha$, then $\alpha\perp\beta$.

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