
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 seven consecutive vertices of a regular 100-gon, checkers of seven different colors are placed. In one move, you are allowed to move any checker 10 positions clockwise to the 11th position if it is free. The goal is to gather the checkers in the seven vertices following the initial vertices. How many distinct arrangements of checkers can be obtained in these seven vertices?

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