
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 $\alpha$, $\beta$, $\gamma$, consider the following propositions:
1. If $a \subset \alpha$ and $b \parallel \alpha$, then $a \parallel b$;
2. If $a \subset \alpha$, $b \subset \beta$, and $\alpha \parallel \beta$, then $a \parallel b$;
3. If $a \parallel \alpha$ and $b \parallel \alpha$, then $a \parallel b$;
4. If $a \perp b$ and $b \parallel c$, then $a \perp c$.

Which of these propositions are correct?

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