
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 the triangle \(ABC\). Consider all tetrahedra \(ABCP\) in which the height from the vertex \(P\) is the smallest among the four altitudes of the tetrahedron. What is the locus of the orthogonal projection of point \(P\) onto the plane \(ABC\)?

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