
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 $a$ and $b$, and two different planes $\alpha$ and $\beta$, determine the correct propositions from the following list:  
① If $\alpha \parallel a$ and $\beta \parallel a$, then $\alpha \parallel \beta$;  
② If $\alpha \perp a$ and $\beta \perp a$, then $\alpha \parallel \beta$;  
③ If $a \parallel \alpha$ and $b \parallel \alpha$, then $a \parallel b$;  
④ If $a \perp \alpha$ and $b \perp \alpha$, then $a \parallel b$.  
Fill in the correct proposition numbers only.

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