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 \[ a - b - c - x - y - z.\]