
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 two different lines $m$, $n$, and two non-coincident planes $\alpha$, $\beta$, the correct statement among the following is ____. (Fill in the serial number)

$①$ If $m \perp \alpha$, $n \perp \beta$, $\alpha \perp \beta$, then $m \perp n$;

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

$③$ If $m \parallel \alpha$, $n \parallel \beta$, $\alpha \parallel \beta$, then $m \parallel n$;

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

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