
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 parallel faces of a cube are $ABCD$ and $A_1B_1C_1D_1$. On the edges $AB, BC, CD, DA; A_1D_1, D_1C_1, C_1B_1, B_1A_1; AA_1, CC_1; B_1B, D_1D$ mark the division point that divides each edge in the ratio $\lambda$. (On an edge $XY$, the division point $Z$ dividing the edge in the ratio $\lambda$ is defined as the internal point for which $XZ : ZY = \lambda$.) Cut off each vertex of the cube with a plane passing through the three division points on the edges originating from that vertex. For which value of $\lambda$ will the remaining volume of the shape be minimized?

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