
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 $a$ and $b$ be two lines, and $\alpha$ and $\beta$ be two planes. Among the following four sets of conditions, all that can deduce $a \perpendicular b$ are \_\_\_\_\_\_\_\_(fill in the serial number).

① $a \subset \alpha$, $b \parallel \beta$, $\alpha \perpendicular \beta$; ② $a \perpendicular \alpha$, $b \perpendicular \beta$, $\alpha \perpendicular \beta$;

③ $a \subset \alpha$, $b \perpendicular \beta$, $\alpha \parallel \beta$; ④ $a \perpendicular \alpha$, $b \parallel \beta$, $\alpha \parallel \beta$.

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