
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 $\alpha$ represent a plane, and $a$, $b$, $c$ represent three different lines. Consider the following five propositions:  
(1) If $a \parallel \alpha$ and $b \parallel \alpha$, then $a \parallel b$.  
(2) If $a \parallel b$ and $b \subset \alpha$, then $a \parallel \alpha$.  
(3) If $a \perp c$ and $b \perp \alpha$, then $a \parallel b$.  
(4) If $a \perp b$, $a \perp c$, $b \subset \alpha$, and $c \subset \alpha$, then $a \perp \alpha$.  
(5) If $a \parallel b$, $b \perp \alpha$, and $c \perp \alpha$, then $a \parallel c$.  
Among these propositions, the correct ones are.

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