
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
In some squares of a  $2012\times 2012$  grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle  $B$  flies to the centre of the square on which it lands is called the *translation vector* of beetle  $B$ . 
For all possible starting and ending configurations, find the maximum length of the sum of the  *translation vectors* of all beetles.

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