
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 distinct lines, and $\alpha$, $\beta$ are distinct planes, select the correct statements from the following (fill in all correct numbers):
① If $a \parallel b$, $b \subset \alpha$, then $a \parallel \alpha$;  
② If $a \perp b$, $a \perp c$, $b \subset \alpha$, $c \subset a$, then $a \perp \alpha$;  
③ If $a \perp \alpha$, $a \subset \beta$, then $\alpha \perp \beta$;  
④ If $a \parallel \beta$, $b \parallel \beta$, $a \subset \alpha$, $b \subset \alpha$, then $\alpha \parallel \beta$.

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