Permutation Distances

\( x_i \) is a permutation of \( \{ 1, 2, \ldots, 402 \} \). What is the largest value of \(S = | \ldots | | |x_1 - x_2| -x_3| - x_4 | - \ldots -x_{402} | \)?

Details and assumptions

\( | \cdot | \) is the absolute value function, which satisfies

\( |x| = \begin{cases} x & x \geq 0 \\ -x & x < 0 \end{cases} \\ \)

For example, \( | 5 | = 5, |-7| = 7 \).

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