
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 $\alpha \perp \beta$, $\alpha \cap \beta = l$, and a point $P \in \alpha$, $P \notin l$, among the following statements, the correct one(s) is/are \_\_\_\_\_\_ . (Fill in the serial number only)  
① A plane through $P$ and perpendicular to $l$ is perpendicular to $\beta$;  
② A line through $P$ and perpendicular to $l$ is perpendicular to $\beta$;  
③ A line through $P$ and perpendicular to $\alpha$ is parallel to $\beta$;  
④ A line through $P$ and perpendicular to $\beta$ lies within $\alpha$.

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