What now? Minimum? Complicating...

Consider all sequences of positive real numbers $x_i$ , $i = 1,2, \ldots, 333$ that satisfy $\sum_{i=1}^{333} x_i = 1$ . The minimum value of

$\sum_{i=1}^{333} \frac{x_i}{1+\sum_{i \neq j, j=1}^{333} x_j}$

is $M$ . Let $M = \frac{a}{b}$ . Find the value of $a+b$ .