
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
Among the following propositions in two mutually perpendicular planes:  
① A known line in one plane must be perpendicular to any line in the other plane;  
② A known line in one plane must be perpendicular to countless lines in the other plane;  
③ Any line in one plane must be perpendicular to countless lines in the other plane;  
④ A line perpendicular to another plane drawn through any point in one plane must lie within the first plane;  
The correct proposition numbers are.

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