
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
In a regular tetrahedron \( A B C D \) with edge length \( a \), points \( A_1, B_1, C_1, D_1 \) are taken on the faces \( B C D, C D A, D A B, A B C \) respectively, such that \( A_1 B_1, B_1 C_1, C_1 D_1, D_1 A_1 \) are perpendicular to the faces \( B C D, C D A, D A B, A B C \) respectively. Find the volume of the tetrahedron \( A_1 B_1 C_1 D_1 \).

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