Minimum Insertions

You are given a sequence of integers $A_1, A_2, \ldots, A_N$. This sequence is circular ― for each valid $i$, the element $A_{i+1}$ follows after $A_i$, and the element $A_1$ follows after $A_N$. You may insert any positive integers at any positions you choose in this sequence; let's denote the resulting sequence by $B$. This sequence is also circular. For each pair of its elements $B_s$ and $B