Solve the system of equations ∣a1−a2∣x2+ ∣a1−a3∣x3+∣a1−a4∣x4=1∣a2−a1∣x1+ ∣a2−a3∣x3+∣a2−a3∣x3=1∣a3−a1∣x1+∣a3−a2∣x2=1∣a4−a1∣x1+∣a4−a2∣x2+ ∣a4−a3∣x3=1\begin{aligned} &|a_1 - a_2| x_2 & +\, |a_1 - a_3| x_3 & + |a_1 - a_4| x_4 &= 1 \\ |a_2 - a_1| x_1 & & +\, |a_2 - a_3| x_3 & + |a_2 - a_3| x_3 &= 1 \\ |a_3 - a_1| x_1 & + |a_3 - a_2| x_2 & & &= 1 \\ |a_4 - a_1| x_1 & + |a_4 - a_2| x_2 & +\, |a_4 - a_3| x_3 & &= 1 \end{aligned}∣a2−a1∣x1∣a3−a1∣x1∣a4−a1∣x1∣a1−a2∣x2+∣a3−a2∣x2+∣a4−a2∣x2+∣a1−a3∣x3+∣a2−a3∣x3+∣a4−a3∣x3+∣a1−a4∣x4+∣a2−a3∣x3=1=1=1=1 where a1,a2,a3,a4a_1, a_2, a_3, a_4a1,a2,a3,a4 are four different real numbers.