CMO 2013 Problem 4

Let $n \geqslant 2$ be an integer. There are $n$ finite sets ${A_1},{A_2},\ldots,{A_n}$ which satisfy the condition $$\left| {{A_i}\Delta {A_j}} \right| = \left| {i - j} \right| \forall i,j \in \left\{ {1,2,...,n} \right\}.$$ Find the minimum of $\sum\limits_{i = 1}^n {\left| {{A_i}} \right|}$ .