
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
A kingdom has  $2021$  towns. All of the towns lie on a circle, and there is a one-way road going from every town to the next  $101$  towns in a clockwise order. Each road is colored in one color. Additionally, it is known that for any ordered pair of towns  $A$  and  $B$  it is possible to find a path from  $A$  to  $B$  so that no two roads of the path would have the same color. Find the minimal number of road colors in the kingdom.

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