
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
Point \( P \) is an interior point of the face \( ABC \) of a regular tetrahedron \( ABCD \). Given the lengths \( PA, PB, \) and \( PC \), calculate the length of the segment \( PD \).

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