<|im_start|>system
You are a helpful assistant. You first think about the reasoning process in the mind and then provide the user with the answer. You should try to truely solve the question by yourself without cheating.  <|im_end|>
<|im_start|>user
 Hydras consist of heads and necks (each neck connects exactly two heads). With one sword strike, all the necks emanating from a head \( A \) of the hydra can be cut off. However, a new neck immediately grows from head \( A \) to every head that \( A \) was not previously connected to. Hercules wins if he manages to split the hydra into two disconnected parts. Find the smallest \( N \) for which Hercules can defeat any 100-headed hydra by making no more than \( N \) strikes. Show your work in <think> </think> tags. And return the final answer in <answer> </answer> tags.<|im_end|>
<|im_start|>assistant
Let me solve this step by step.
<think>
user
10
Hydras consist of heads and necks (each neck connects exactly two heads). With one sword strike, all the necks emanating from a head \( A \) of the hydra can be cut off. However, a new neck immediately grows from head \( A \) to every head that \( A \) was not previously connected to. Hercules wins if he manages to split the hydra into two disconnected parts. Find the smallest \( N \) for which Hercules can defeat any 100-headed hydra by making no more than \( N \) strikes.
test