
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 in space, and $\alpha$ and $\beta$ be two different planes in space. Below are four propositions:  
1. If $m \perp \alpha$, $n \parallel \beta$, and $\alpha \parallel \beta$, then $m \perp n$;  
2. If $m \perp n$, $\alpha \parallel \beta$, and $m \perp \alpha$, then $n \parallel \beta$;  
3. If $m \perp n$, $\alpha \parallel \beta$, and $m \parallel \alpha$, then $n \perp \beta$;  
4. If $m \perp \alpha$, $m \nparallel n$, and $\alpha \parallel \beta$, then $n \perp \beta$;  
Among these, the numbers of the true propositions are __________ (write down all the numbers of the true propositions).

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