Korean Mathematical Olympiad

For three mutually distinct real numbers $a_1$, $a_2$, $a_3$, define three real numbers $b_1$, $b_2$, $b_3$ as follows: $$b_j=\left(1+\frac{a_j{a}_i}{a_j-a_i}\right)\left(1+\frac{a_j{a}_k}{a_j-a_k}\right),\{i,j,k\}=\{1,2,3\}.$$ Prove the inequality $$1+|a_1{b}_1+a_2{b}_2+a_3{b}_3|\le (1+|a_1|)(1+|a_2|)(1+|a_3|).$$ When does the equality hold?