LCM New Year

Number Theory Level 3

Consider all possible natural numbers $a, b, c, x, y,$ and $z$ such that
1) $LCM(a,b,c) = LCM(x,y,z) = 2015,$
2) both $a+b+c$ and $x+y+z$ are perfect squares,
3) $a, b, c$ are distinct,
4) $x, y, z$ are distinct.

Find the maximum value of $a - b - c - x - y - z.$