
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 unit cube is cut along two sets of parallel planes, one horizontal and one vertical, dividing the cube into smaller pieces. The first set of cuts is made halfway up the cube creating a horizontal division, and the second set is made along two vertical planes each passing through the midpoint of opposite faces of the cube. What is the volume of the piece that contains vertex $A$ of the original cube?

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