Disjoint Segment Sums 2000 pts

Determine all integers $n\ge 2$ for which there are two sequences $a_1,\ldots,a_n$ and $b_1,\ldots,b_n$ of positive integers such that:

• Each sequence contains at least two terms with different values.
• No nonempty contiguous segment of one sequence has the same sum as a nonempty contiguous segment of the other.
• Their combined total is

$$\sum_{i=1}^{n}(a_i+b_i)=n(n+2).$$