
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

Chichikov is playing a game with Nozdryov. Initially, Nozdryov distributes 1001 nuts into three boxes. After observing the distribution, Chichikov names any integer \( N \) from 1 to 1001. Then, Nozdryov must move, if necessary, one or more nuts to an empty fourth box and present Chichikov with one or more boxes that contain exactly \( N \) nuts in total. As a result, Chichikov will receive as many dead souls as the number of nuts Nozdryov moved. What is the maximum number of dead souls Chichikov can guarantee to get, regardless of how Nozdryov distributes the nuts?

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