Školjka

%V0 Let $n$ be a positive integer, and consider a sequence $a_1, a_2, \ldots, a_n$ of positive integers. Extend it periodically to an infinite sequence $a_1, a_2, \ldots$ by defining $a_{n+i} = a_i$ for all $i \geq 1$. If $$ a_1 \leq a_2 \leq \cdots \leq a_n \leq a_1 + n $$ and $$ a_{a_i} \leq n + i - 1 \quad \quad \text{for}\ i = 1, 2, \ldots, n \text{,} $$ prove that $$ a_i + \cdots + a_n \leq n^2 \text{.} $$