
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

The planar angle at the vertex of a regular triangular pyramid \( A B C D \) with a base \( A B C \) is \( \alpha \). A regular truncated pyramid \( A B C A_1 B_1 C_1 \) is cut along the five edges: \( A_1 B_1 \), \( B_1 C_1 \), \( C_1 C \), \( C A \), and \( A B \). Then this pyramid is unfolded onto a plane. For what values of \( \alpha \) will the resulting development necessarily overlap itself?

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