Question 20
Let $a_1,a_2,a_3$ be three distinct integers where $1000>a_1>a_2>a_3>0$. Suppose there exist real numbers $x,y,z$ such that $$\begin{aligned}(a_1-a_2)y+(a_1-a_3)z&=a_1+a_2+a_3,\\(a_1-a_2)x+(a_2-a_3)z&=a_1+a_2+a_3,\\(a_1-a_3)x+(a_2-a_3)y&=a_1+a_2+a_3.\end{aligned}$$ Find the largest possible value of $x+y+z$.