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If a, b, and c are positive integers less than 13 such that
2*a*b + b*c + c*a == 0 (mod 13)
a*b + 2*b*c + c*a == 4*a*b*c (mod 13)
a*b + b*c + 2*c*a == 8*a*b*c (mod 13)
then determine the remainder when a+b+c is divided by 13.