CMO 1997 Problem 3

Prove that there are infinitely many natural numbers $n$ such that we can divide $1,2,\ldots ,3n$ into three sequences $(a_n),(b_n)$ and $(c_n)$ , with $n$ terms in each, satisfying the following conditions: i) $a_1+b_1+c_1= a_2+b_2+c_2=\ldots =a_n+b_n+c_n$ and $a_1+b_1+c_1$ is divisible by $6$ ; ii) $a_1+a_2+\ldots +a_n= b_1+b_2+\ldots +b_n=c_1+c_2+\ldots +c_n,$ and $a_1+a_2+\ldots +a_n$ is divisible by $6$ .