
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
Twelve congruent spheres are arranged inside a unit cube such that one sphere is located at one vertex of the cube and each of the remaining eleven spheres is tangent to this vertex sphere and to three faces of the cube. What is the radius of each sphere?
A) $\frac{1}{3}$
B) $\frac{1}{4}$
C) $\frac{1}{2}$
D) $\frac{1}{6}$
E) $\frac{2}{3}$

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