Nemotron-IMO-Bench 2026 N35

Prove that for any integer $n>1$ there are only finitely many $n$-tuples of positive integers $(a_1,a_2,...,a_n)$ such that $a_1>a_2>...>a_n$, $\gcd(a_1,a_2,...,a_n)=1$ and $$a_1=\gcd(a_1,a_2)+\gcd(a_2,a_3)+...+\gcd(a_{n-1},a_n)+\gcd(a_n,a_1).$$