Almost 0

Consider a \( 2017 \times 2017 \) square grid. Each cell is filled in with a real number with absolute value at most 1, such that the sum of any \( 2 \times 2 \) square grid is exactly 0.

Let \(S\) be the sum of all of these real numbers. What is the maximum value of \(S\)?

Note: Below is a \(3 \times 3\) grid which satisfies the requirements above (although its sum is not necessarily maximized).

\[\Large \begin{array}{|c|c|c|} \hline -0.3 & 0.2 & -0.6 \\ \hline 0.4 & -0.3 & 0.7 \\ \hline 0.5 & -0.6 & 0.2 \\ \hline \end{array}\]

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