
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
For four distinct points A, B, C, and D in space, consider the following five propositions:   
1. If lines AB and CD are coplanar, then lines AC and BD are coplanar as well.  
2. If lines AB and CD are skew lines (not coplanar), then lines AC and BD are skew lines as well.  
3. If $AB=AC$ and $DB=DC$, then $AD \perp BC$.  
4. If $AB \perp CD$ and $AC \perp BD$, then $AD \perp BC$.  
5. If $AB=AC=AD$ and $BC=CD=DB$, then points A, B, C, D must be the vertices of a regular tetrahedron.  
Which of the above propositions are correct? Fill in all the correct proposition numbers.

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