
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 five propositions about the tetrahedron P-ABC, the correct ones are ______. (1) If triangle ABC is an equilateral triangle and the dihedral angles between the lateral faces and the base are all equal, then the tetrahedron P-ABC is a regular tetrahedron; (2) If triangle ABC is an equilateral triangle and all lateral faces are isosceles triangles, then the tetrahedron P-ABC is a regular tetrahedron; (3) If triangle ABC is an equilateral triangle, and the projection of point A on the lateral face PBC is the orthocenter of triangle PBC, then P-ABC is a regular tetrahedron; (4) If all the edges of tetrahedron P-ABC are equal, then the ratio of the radius of the circumscribed sphere to the height is 3:4; (5) If all the edge lengths of tetrahedron P-ABC are equal, and a point M moves within lateral face PAB such that the distance of point M from face ABC is equal to its distance from point P, then M's trajectory is part of an ellipse.

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