
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
Player  $A$  places an odd number of boxes around a circle and distributes  $2013$  balls into some of these boxes. Then the player  $B$  chooses one of these boxes and takes the balls in it. After that the player  $A$  chooses half of the remaining boxes such that none of two are consecutive and take the balls in them. If player  $A$  guarantees to take  $k$  balls, find the maximum possible value of  $k$ .

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