
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
Given is a cube of side length  $2021$ . In how many different ways is it possible to add somewhere on the boundary of this cube a  $1\times 1\times 1$  cube in such a way that the new shape can be filled in with  $1\times 1\times k$  shapes, for some natural number  $k$ ,  $k\geq 2$ ?

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