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

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 $m$