
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
Alexis imagines a  $2008\times 2008$  grid of integers arranged sequentially in the following way:

\[\begin{array}{r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r@{\hspace{20pt}}r}1,&2,&3,&\ldots,&20082009,&2010,&2011,&\ldots,&40264017,&4018,&4019,&\ldots,&6024\vdots&&&&\vdots2008^2-2008+1,&2008^2-2008+2,&2008^2-2008+3,&\ldots,&2008^2\end{array}\]
She picks one number from each row so that no two numbers she picks are in the same column.  She them proceeds to add them together and finds that  $S$  is the sum.  Next, she picks  $2008$  of the numbers that are distinct from the  $2008$  she picked the first time.  Again she picks exactly one number from each row and column, and again the sum of all  $2008$  numbers is  $S$ .  Find the remainder when  $S$  is divided by  $2008$ .

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