Chef, Chefina and Restoration

Chefina gave Chef a sequence $F_1, F_2, \ldots, F_N$ as a birthday present. She constructed it from a sequence of positive integers $A_1, A_2, \ldots, A_N$ in the following way: for each valid $i$, $F_i = \sum_{j=1}^i (-1)^{(j+1)} \cdot A_j$. Unfortunately, Chef was playing with the sequence $F$ and shuffled it. Now, he has a sequence denoted by $B$, but he does not remember $F$. The only thing