
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 distinct lines $a$ and $b$, and two distinct planes $\alpha$ and $\beta$, consider the following four propositions:
1. If $a \perp b$, $a \perp \alpha$, and $b \nsubseteq \alpha$, then $b \parallel \alpha$;
2. If $a \parallel \alpha$, $\alpha \perp \beta$, then $a \perp \beta$;
3. If $a \perp \beta$, $\alpha \perp \beta$, then $a \parallel \alpha$ or $a \subseteq \alpha$;
4. If $a \perp b$, $a \perp \alpha$, $b \perp \beta$, then $\alpha \perp \beta$.

Which of the above propositions are correct? Fill in all the correct proposition numbers.

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