
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 an infinite checkered paper with a cell side of one unit. The distance between two cells is defined as the length of the shortest path of a rook from one cell to another (considering the path from the center of the rook). What is the minimum number of colors needed to color the board (each cell is to be painted with one color) such that two cells that are a distance of 6 apart are always painted different colors?

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