1. On the board, the numbers $a, b$, and $c$ are written. They were erased, and in their place, the numbers $a^{2}+2 b c, b^{2}+2 c a$, $c^{2}+2 a b$ were written. After this, it turned out that the numbers on the board are the same as those at the beginning (possibly in a different order). Find all possible values of the sum $a+b+c$.

$(25$ points. $)$