
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 with Nozdryov. Initially, Nozdryov distributes 1001 nuts into three boxes. After viewing the distribution, Chichikov names any integer \( N \) from 1 to 1001. Then, Nozdryov must (if necessary) move one or more nuts to an empty fourth box and present to Chichikov one or more boxes containing a total of exactly \( N \) nuts. 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 himself, regardless of Nozdryov's strategy?

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