
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
In the image, the plane is covered with hexagons of seven colors such that the centers of hexagons of the same color form the vertices of a grid of identical equilateral triangles. For what number \( n \) of colors is a similar construction possible?

Note: Only such fillings of the plane with shapes (squares or hexagons) are considered where the grid corresponding to any one color has the same sizes and directions of sides of squares (or triangles) as the grid corresponding to any other color (that is, all grids must be obtained from each other by parallel shift).

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