
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 integer array consisting of 121 integers, each ranging from 1 to 1000 (duplicates allowed), this array has a unique mode (the number that appears the most frequently). Let \( D \) be the difference between this mode and the arithmetic mean of the array. When \( D \) reaches its maximum value, what is the integer part of \( D \)?

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