
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 one wolf disguised as a sheep in a field. Some sheep are friends with each other (friendship is mutual). The wolf's goal is to eat all the sheep. Initially, the wolf chooses some sheep to become friends with. Each following day, the wolf eats one of its friends. Each time the wolf eats a sheep $A$: (a) If a friend of $A$ was originally a friend, they are no longer friends with the wolf. (b) If a friend of $A$ was not originally a friend of the wolf, they become friends with the wolf. The wolf repeats this procedure until it has eaten as many sheep as possible.

Find the largest integer $m$ as a function of $n$ satisfying the following condition: there exists a choice of friendships among the sheep such that there are exactly $m$ ways to choose the initially friendly sheep, allowing the wolf to eat all the sheep.

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