
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 tetrahedron $ABCD$, the following propositions are given:  
① If $AC \perp BD$, and $AB \perp CD$, then $AD \perp BC$;  
② If $E$, $F$, and $G$ are the midpoints of $BC$, $AB$, and $CD$ respectively, then the size of $\angle FEG$ is equal to the angle formed by the skew lines $AC$ and $BD$;  
③ If point $O$ is the center of the circumscribed sphere of tetrahedron $ABCD$, then the projection of $O$ on plane $ABD$ is the circumcenter of $\triangle ABD$;  
④ If all four faces are congruent triangles, then the tetrahedron $ABCD$ is a regular tetrahedron.  
Among these propositions, the correct ones are (fill in all the correct proposition numbers).

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