\(\mathfrak{F}(x)=\dfrac{1}{x-1}+\dfrac{1}{x-2}+\dfrac{1}{x-3} + \ldots +\dfrac{1}{x-99}+\dfrac{1}{x-100}\)

Given the above function \(\mathfrak{F}(x) \) and

\(\large x_{1},x_{2}, \ldots ,x_{n}\) are its roots in some order.

Find the maximum possible value of value

\[\large\dfrac{ [x_{1} ]-[ x_{2} ]+ [ x_{3} ]-[ x_{4} ]+ \ldots \pm[ x_{n} ]} {n+1}\]

**Details and Assumptions**

\([m]\) represents greatest integer function of \(m\)

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