
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 $b$ and $c$ represent two lines, and $\alpha$ and $\beta$ represent two planes. The following statements are given:  
① If $b \subset \alpha$ and $c \parallel \alpha$, then $b \parallel c$;  
② If $b \subset \alpha$ and $b \parallel c$, then $c \parallel \alpha$;  
③ If $c \parallel \alpha$ and $\alpha \perp \beta$, then $c \perp \beta$;  
④ If $c \parallel \alpha$ and $c \perp \beta$, then $\alpha \perp \beta$.  
Among these statements, the correct ones are . (Write down the numbers of all correct statements)

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