
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
2009 whole non-negative numbers, each not exceeding 100, are arranged in a circle. It is allowed to add 1 to two adjacent numbers, with the restriction that any pair of adjacent numbers can undergo this operation no more than $k$ times. What is the smallest value of $k$ such that all numbers can be made equal?

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