
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 $α⊥β$, $α∩β=l$, point $P∈α$, and $P∉l$, the correct proposition among the following is $\_\_\_\_\_\_$ (only fill in the number).
$①$ The plane passing through $P$ and perpendicular to $l$ is perpendicular to $β$;
$②$ The line passing through $P$ and perpendicular to $l$ is perpendicular to $β$;
$③$ The line passing through $P$ and perpendicular to $α$ is parallel to $β$;
$④$ The line passing through $P$ and perpendicular to $β$ is within $α$.

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