
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 that $a$, $b$, $c$ are lines, and $\beta$ is a plane, consider the following propositions:  
① If $a \perp b$ and $b \perp c$, then $a \parallel c$;  
② If $a \parallel b$ and $b \perp c$, then $a \perp c$;  
③ If $a \parallel \beta$, $a \subset \alpha$, and $\alpha \cap \beta = b$ then $a \parallel b$;  
④ If $a$ and $b$ are skew lines, and $a \parallel \beta$, then $b$ intersects $\beta$;  
Among these, the numbers of the true propositions are. (You need to list all the numbers of the true propositions)

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