
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 tetrahedrons \(A B C D\) and \(A_{1} B_{1} C_{1} D_{1}\) are positioned such that all vertices of each of them lie in the corresponding planes of the faces of the other tetrahedron (for example, \(A\) lies in the plane \(B_{1} C_{1} D_{1}\) and so on). Moreover, \(A_{1}\) coincides with the centroid of the triangle \(B C D\), and the lines \(B D_{1}, C B_{1},\) and \(D C_{1}\) bisect the segments \(A C, A D,\) and \(A B\) respectively. Find the volume of the intersection of the two tetrahedrons, given that the volume of the tetrahedron \(A B C D\) is \(V\).

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