
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 two different lines, and $\alpha$ and $\beta$ are two different planes, the following four propositions are given:

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

Among these, the correct proposition numbers are __________.

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