
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 positive integer  $n$  is given. A cube  $3\times3\times3$  is built from  $26$  white and  $1$  black cubes  $1\times1\times1$  such that the black cube is in the center of  $3\times3\times3$ -cube. A cube  $3n\times 3n\times 3n$  is formed by  $n^3$  such  $3\times3\times3$ -cubes. What is the smallest number of white cubes which should be colored in red in such a way that every white cube will have at least one common vertex with a red one.

<details><summary>thanks</summary>Thanks to the user Vlados021 for translating the problem.</details>

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