
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 a $2009 \times 2009$ grid, each small square is filled with a real number whose absolute value does not exceed 1. In addition, for every $2 \times 2$ square, the sum of the 4 real numbers is 0. Find the maximum sum of all the numbers in the grid.

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