
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
Find the smallest positive integer $N$ such that any "hydra" with 100 necks, where each neck connects two heads, can be defeated by cutting at most $N$ strikes. Here, one strike can sever all the necks connected to a particular head $A$, and immediately after, $A$ grows new necks to connect with all previously unconnected heads (each head connects to one neck). The hydra is considered defeated when it is divided into two disconnected parts.

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