
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
A point $O$ is located on the segment connecting the vertex of a triangular pyramid with the point of intersection of the medians of the base. Find the volume of the common part of the given pyramid and a pyramid symmetrical to it with respect to point $O$, given that point $O$ divides the aforementioned segment in the ratio of: a) $1:1$; b) $3:1$; c) $2:1$; d) $4:1$ (counting from the vertex).

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