# 4-tuples with close sums

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 $$a_i,b_i,c_i,d_i\leq 1$$. Is it always possible to partition $$\{1,2,\dots,n\}$$ into two subsets $$X,Y$$ so that at least $$3$$ of the following $$4$$ inequalities hold:

$$\left|\sum_Xa_i-\sum_Ya_i\right|\leq 1, \left|\sum_Xb_i-\sum_Yb_i\right|\leq 1$$ $$\left|\sum_Xc_i-\sum_Yc_i\right|\leq 1, \left|\sum_Xd_i-\sum_Yd_i\right|\leq 1$$

An affirmative answer here would imply one for this (unsolved) question.