
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 lines $a$, $b$ and planes $\alpha$, $\beta$, which of the following statements are correct? (Write down the numbers of all correct statements)  
① If $\alpha \parallel \beta$ and $a \parallel \alpha$, then $a \parallel \beta$;  
② If $a \perp b$, $a \perp \alpha$, and $b \perp \beta$, then $\alpha \perp \beta$;  
③ If $\alpha \perp \beta$ and $a \perp \beta$, then $a \parallel \alpha$;  
④ If $a \parallel \alpha$ and $a \perp \beta$, then $\alpha \perp \beta$.

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