
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
There are  $n$  sheep and a wolf in sheep's clothing . Some of the sheep are friends (friendship is mutual). The goal of the wolf is to eat all the sheep. First, the wolf chooses some sheep to make friend's with. In each of the following days, the wolf eats one of its friends. Whenever the wolf eats a sheep  $A$ :
(a) If a friend of  $A$  is originally a friend of the wolf, it un-friends the wolf.
(b) If a friend of  $A$  is originally not a friend of the wolf, it becomes a friend of the wolf.
Repeat the procedure until the wolf has no friend left.
Find the largest integer  $m$  in terms of  $n$  satisfying the following: There exists an initial friendsheep structure such that the wolf has  $m$  different ways of choosing initial sheep to become friends, so that the wolf has a way to eat all of the sheep.

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