<|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
 A hydra consists of several heads and several necks, where each neck joins two heads.  When a hydra's head  $A$  is hit by a sword, all the necks from head  $A$  disappear, but new necks grow up to connect head  $A$  to all the heads which weren't connected to  $A$ .  Heracle defeats a hydra by cutting it into two parts which are no joined.  Find the minimum  $N$  for which Heracle can defeat any hydra with  $100$  necks by no more than  $N$  hits. 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
A hydra consists of several heads and several necks, where each neck joins two heads.  When a hydra's head  $A$  is hit by a sword, all the necks from head  $A$  disappear, but new necks grow up to connect head  $A$  to all the heads which weren't connected to  $A$ .  Heracle defeats a hydra by cutting it into two parts which are no joined.  Find the minimum  $N$  for which Heracle can defeat any hydra with  $100$  necks by no more than  $N$  hits.
test