
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
Consider an arbitrary convex polyhedron and the number $\kappa=V^{2} / F^{3}$, where $V$ is the volume of the polyhedron and $F$ is its surface area. How does this number change as we proceed through the following 5 polyhedra:

1. the regular tetrahedron $ABCD$;
2. a new polyhedron obtained by cutting off the tetrahedron $AA_{1}A_{2}A_{3}$ from it, where $A_{1}, A_{2}, A_{3}$ are the points on edges $AB, AC, AD$ respectively, such that $AA_{1} / AB=AA_{2} / AC=AA_{3} / AD=\lambda \leq 1 / 2$;

3 - 5. subsequently cut off the remaining similar tetrahedra $BB_{1}B_{2}B_{3}$, $CC_{1}C_{2}C_{3}$, and finally $DD_{1}D_{2}D_{3}$ in the same manner.

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