
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
Two identical triangular pyramids with volume $V$ are placed in space symmetrically about point $O$. Find the volume of their common part if point $O$ lies on the segment connecting the vertex of the pyramid to the centroid of the base, and divides this segment in the ratio 1) $1:1$, 2) $3:1$, 3) $2:1$, 4) $4:1$ from the vertex.

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