
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 $m$ and $n$ are two different lines, and $\alpha$ and $\beta$ are two non-coincident planes, the correct proposition is ______ (fill in the serial number).

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

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