
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 points \(A, B, C, D, E,\) and \(F\) are the vertices of the lower base of a right hexagonal prism, the points \(M, N, P, Q, R,\) and \(S\) are the midpoints of the sides of the upper base, and the points \(O\) and \(O_1\) are the centers of the lower and upper bases, respectively. Find the volume of the intersection of pyramids \(O_1 ABCDEF\) and \(O MN PQRS\) if the volume of the prism is \(V\).

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