KöMaL 2018 B4980

Let $n>3$ be a positive integer, and let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove that $1 < \frac{a_1}{a_n+a_1+a_2} + \frac{a_2}{a_1+a_2+a_3} + \ldots + \frac{a_n}{a_{n-1}+a_n+a_1} < \left[ \frac{n}{2} \right]$ where the left-hand side of the inequality cannot be replaced by a larger number, and the right-hand side cannot be replaced by a smaller number. ($[x]$ denotes the greatest integer not greater than the number $x$.)