
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
Given point P outside the plane α containing triangle ABC, let O be the orthogonal projection of point P onto plane α.
(1) If the distances from P to the three vertices of triangle ABC are equal, then O is the circumcenter of triangle ABC;
(2) If the angles formed by PA, PB, PC and the plane α are equal, then O is the incenter of triangle ABC;
(3) If the distances from P to the three sides of triangle ABC are equal and O is inside triangle ABC, then O is the incenter of triangle ABC;
(4) If the angles formed by the planes PAB, PBC, PCA and plane α are equal and O is inside triangle ABC, then O is the circumcenter of triangle ABC;
(5) If PA, PB, and PC are pairwise perpendicular, then O is the orthocenter of triangle ABC.
Write down the numbers of the statements you believe to be correct.

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