PE-MO 2013 Problem 2

The positive integers $a, b, c$ are such that $$gcd \,\, (a, b, c) = 1,$$ $$gcd \,\,\,(a, b + c) > 1,$$ $$gcd \,\,\,(b, c + a) > 1,$$ $$gcd \,\,\,(c, a + b) > 1.$$ Determine the smallest possible value of $a + b + c$ .

Clarification: gcd stands for greatest common divisor.