
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
Let \( A B C D \) be a regular tetrahedron and \(\mathcal{Z}\) an isometry mapping \( A, B, C, D \) into \( B, C, D, A \), respectively. Find the set \(\mathcal{M}\) of all points \( X \) of the face \( A B C \) whose distance from \(\mathcal{Z}(X)\) is equal to a given number \( t \). Find necessary and sufficient conditions for the set \(\mathcal{M}\) to be nonempty.

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