
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 the lines $a$, $b$, $c$, and the planes $\alpha$, $\beta$, the following propositions are provided:  
(1) If $a \parallel \alpha$, $b \parallel \beta$, then $a \parallel b$; (2) If $a \parallel \alpha$, $b \perp \alpha$, then $a \perp b$;  
(3) If $a \parallel b$, $b \parallel \alpha$, then $a \parallel \alpha$; (4) If $a \perp \alpha$, $a \parallel \beta$, then $\alpha \perp \beta$.  
Among these, the correct propositions are (fill in the correct proposition numbers).

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