
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
Players \( A \) and \( B \) play a game with \( N \geq 2012 \) coins and 2012 boxes arranged around a circle. Initially, \( A \) distributes the coins among the boxes so that there is at least 1 coin in each box. Then the two of them make moves in the order \( B, A, B, A, \ldots \) by the following rules:
- On every move of his, \( B \) passes 1 coin from every box to an adjacent box.
- On every move of hers, \( A \) chooses several coins that were not involved in \( B \)'s previous move and are in different boxes. She passes every chosen coin to an adjacent box.

Player \( A \)'s goal is to ensure at least 1 coin in each box after every move of hers, regardless of how \( B \) plays and how many moves are made. Find the least \( N \) that enables her to succeed.

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