
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 vertices of a regular 100-gon, 100 chips numbered $1, 2, \ldots, 100$ are placed in this order clockwise. In one move, it is allowed to swap two chips located in adjacent vertices if the numbers on these chips differ by no more than $k$. What is the smallest value of $k$ such that through a series of these moves, it is possible to arrange the chips so that each chip is shifted one position clockwise (relative to its initial position)?

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