KURSCHAK 1986 Problem 2

Let $n>2$ be a positive integer. Find the largest value $h$ and the smallest value $H$ for which $$h<{a_1\over a_1+a_2}+{a_2\over a_2+a_3}+\cdots+{a_n\over a_n+a_1}<H$$ holds for any positive reals $a_1,\dots,a_n$ .