
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:**(a)** On every move of his  $B$  passes  $1$  coin from every box to an adjacent box.**(b)** 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 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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