
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
Let  $ n \geq 3$  be an odd integer. We denote by  $ [\minus{}n,n]$  the set of all integers greater or equal than  $ \minus{}n$  and less or equal than  $ n$ .
Player  $ A$  chooses an arbitrary positive integer  $ k$ , then player  $ B$  picks a subset of  $ k$  (distinct) elements from  $ [\minus{}n,n]$ . Let this subset be  $ S$ .
If all numbers in  $ [\minus{}n,n]$  can be written as the sum of exactly  $ n$  distinct elements of  $ S$ , then player  $ A$  wins the game. If not,  $ B$  wins.
Find the least value of  $ k$  such that player  $ A$  can always win the game.

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