
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
Let $m$ and $n$ be two different lines, and $\alpha$, $\beta$, $\gamma$ be three different planes. Among the following propositions, the correct ones are ______. (Write down the numbers of all correct propositions)

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

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