
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 has each edge of 1 dm. On each face of the cube we draw a smaller square, whose center coincides with the center of the face and whose sides are parallel to the edges of the face. The side length of the drawn square is \(a\) on the top and bottom faces, \(b\) on the front and back faces, and \(c\) on the left and right faces, where \(c < b < a < 1\). We successively remove the square prisms defined by the squares with side lengths \(c\), \(b\), and \(a\) (respectively, in the second and third cases, the prisms already removed in previous steps stop affecting the next steps). Determine the values of \(c\), \(b\), and \(a\) such that the volumes of the consecutively removed parts and the volume of the remaining body are equal.

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