
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
The following are four propositions:  
① If two planes are perpendicular, then any line within one of the planes is perpendicular to countless lines within the other plane;  
② Let $m$ and $n$ be two different lines, and $\alpha$ and $\beta$ be two different planes. If $\alpha \parallel \beta$, $m \perpendicular \alpha$, and $n \parallel \beta$, then $m \perpendicular n$;  
③ The sufficient but not necessary condition for "line $a \perpendicular b$" is "a is perpendicular to the projection of $b$ in plane $\alpha$";  
④ If point $P$ is equidistant from the three sides of a triangle, then the projection of point $P$ onto the plane of the triangle is the incenter of the triangle.  
Among these, the correct proposition numbers are.

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