
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
Let  $n$  be a positive integer. A regular hexagon with side length  $n$  is divided into equilateral triangles with side length  $1$  by lines parallel to its sides. 
Find the number of regular hexagons all of whose vertices are among the vertices of those equilateral triangles.

*UK - Sahl Khan*

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