For $a_i>0(i=1,2,3\dots ,n)$ , Prove: $$ \frac {(a_1+a_2)(a_2+a_3) \dots (a_{n-1}+a_n)(a_1+a_2+a_3+\dots+a_n)}{a_1 a_2 \dots a_n} \ge \left(\frac{\sin{\frac{2\pi}{n+2}}}{\sin{\frac{\pi}{n+2}}}\right)^{n+2} $$
This problem was proposed by Nairi Sedrakyan in IOM (International Olympiad of Metropolises) shortlist 2020, it’s problem A2.
(Fan inequality) For $b_i \in \mathbb{R}\quad(i=1,2,3\dots,n)$ and $\displaystyle\sum_{i=1}^n b_i=0$ ,we have $$ \sum_{i=1}^n b_ib_{i+1} \le \cos{\frac{2\pi}{n}}\sum_{i=1}^n b_i^2 $$
With sine function involved, the first thing I thought was that this was a Fan-Type inequality. For this type, the general method is to use AM-GM or Cauchy inequality with undetermined coefficients to "generate" sine or cosine functions, but it is hard to use either method on LHS.
Another thought is to prove this problem with Lagrange Multiplier Method which can solve Fan inequality. For $n$ variable case, it is hard to solve $n$ equations, so I tried LMM when $n=3$(because $n=2$ is trivial). For this case, the equations can be $$ a_1+a_2+a_3=0\\ a_1a_2a_3=0 $$ which is impossible because of $a_i>0(i=1,2,3\dots ,n)$.
Is there any solution or idea to solve this? Help me, please.