
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 solid cube with edges of length 2 units is cut three times creating several smaller polyhedrons. The first cut divides the cube through one of its faces, intersecting the opposite face diagonally from one vertex to the midpoint of the opposing edge. The second cut also goes through a face, parallel to the first cut, but shifts its intersection point with the opposing face to one-third from the nearest vertex. The third cut is parallel to these two cuts and passes through the two faces, moving the edge from the midpoint to one-quarter of the width along an adjacent edge. What is the volume of the pyramid-shaped section that includes the vertex A (the original vertex of the cube before any cuts were made)?

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