BALTIC-WAY 2002 Problem 1

Solve the system of equations

$$\left\{\begin{array}{l} a^{3}+3 a b^{2}+3 a c^{2}-6 a b c=1 \\ b^{3}+3 b a^{2}+3 b c^{2}-6 a b c=1 \\ c^{3}+3 c a^{2}+3 c b^{2}-6 a b c=1 \end{array}\right.$$

in real numbers.