
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 plays with Nozdryov. Initially, Nozdryov distributes 1001 nuts into three boxes. After looking at the arrangement, Chichikov names any integer \( N \) from 1 to 1001. Then Nozdryov must, if necessary, transfer one or more nuts to an empty fourth box and present Chichikov with one or more boxes containing exactly \( N \) nuts in total. As a result, Chichikov will receive as many dead souls as the number of nuts Nozdryov transferred. What is the maximum number of souls Chichikov can guarantee for himself, regardless of Nozdryov's actions?

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