IR-TST 2018 Problem 12

$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation $b_1,b_2,\ldots,b_n$ of $a_i$ 's such that all the $n$ sums $b_i+b_{i+1}$ are distinct ( $1\le i\le n$ , $b_{n+1}\equiv b_1$ )

Proposed by Mohsen Jamali