
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 $a$, $b$ and planes $\alpha$, $\beta$, among the following propositions, the false ones are (fill in the serial number only).

① If $a \parallel b$, then $a$ is parallel to any plane that passes through $b$;  
② If $a \parallel \alpha$, $b \parallel \alpha$, then $a \parallel b$;  
③ If $a \parallel \alpha$, $b \parallel \beta$, and $\alpha \perp \beta$, then $a \perp b$;  
④ If $\alpha \cap \beta = a$, and $b \parallel \alpha$, then $b \parallel a$.

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