
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 lines $a$ and $b$, and a plane $\alpha$, there are four propositions:  
(1) If $a \parallel b$ and $a \parallel \alpha$ then $b \parallel \alpha$;  
(2) If $a \perp \alpha$ and $b \perp \alpha$ then $a \parallel b$;  
(3) If $a \perp \alpha$ and $a \perp b$ then $b \parallel \alpha$;  
(4) If $a \parallel \alpha$ and $a \perp b$ then $b \perp \alpha$;  
Among these, the correct propositions are signified by their numbers.

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