
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 different lines, and $\alpha$ and $\beta$ be two different planes. Then the following four propositions are given:  
① If $a \perp b$, $a \perp \alpha$, then $b$ is parallel to $\alpha$;  
② If $a$ is parallel to $\alpha$, $\alpha \perp \beta$, then $a \perp \beta$;  
③ If $a \perp \beta$, $\alpha \perp \beta$, then $a$ is parallel to $\alpha$;  
④ If $a \perp b$, $a \perp \alpha$, $b \perp \beta$, then $\alpha \perp \beta$.  
The correct proposition numbers are ____.

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