Some $4$-tuples of positive real numbers $(a_1,b_1,c_1,d_1),\dots,(a_n,b_n,c_n,d_n)$ are given, with all $a_i,b_i,c_i,d_i\leq 1$. Can we always partition $\{1,2,\dots,n\}$ into two subsets $X,Y$ so that $$1+\sum_Xa_i\geq \sum_Ya_i\text{ and } 1+\sum_Xb_i\geq \sum_Yb_i$$ and $$\sum_Xc_i\leq 1+\sum_Yc_i\text{ and } \sum_Xd_i\leq 1+\sum_Yd_i?$$

It is shown here that the partition is possible if we replace the $1$'s by $3$'s. What is the best possible value between $1$ and $3$?