
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
In space, the following four propositions are given:  
(1) Through a point, there is exactly one plane perpendicular to a given line;  
(2) If the distances from two points outside a plane to the plane are equal, then the line passing through these two points must be parallel to the plane;  
(3) The projections of two intersecting lines on the same plane must be intersecting lines;  
(4) In two mutually perpendicular planes, any line in one plane must be perpendicular to countless lines in the other plane.  
Among these propositions, the correct ones are.

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