
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 lines $a$, $b$, $c$ and planes $\alpha$, $\beta$, consider the following propositions:
1. If $a \parallel \alpha$ and $b \parallel \alpha$, then $a \parallel b$;
2. If $\alpha \parallel \beta$ and $c \perp \alpha$, then $c \perp \beta$;
3. If $a \perp b$ and $a \perp \alpha$, then $b \parallel \alpha$;
4. If $\alpha \perp \beta$ and $a \parallel \alpha$, then $a \perp \beta$;
5. If $a \perp c$ and $b \perp c$, then $a \parallel b$ or $a$ and $b$ are skew or $a$ and $b$ intersect;

Choose the correct propositions from the list above (Include all correct propositions' numbers).

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