
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
For the line $m$, line $n$, and the planes $\alpha$, $\beta$, and $\gamma$, consider the following four propositions:

① If $m\parallel\alpha$ and $m\perp n$, then $n\perp\alpha$;

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

③ If $\alpha\perp\beta$ and $\gamma\perp\beta$, then $\alpha\parallel\gamma$;

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

The number(s) of the correct proposition(s) is __________.

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