HMMT 2020 T5

Let $a_{0}, b_{0}, c_{0}, a, b, c$ be integers such that $\operatorname{gcd}\left(a_{0}, b_{0}, c_{0}\right)=\operatorname{gcd}(a, b, c)=1$. Prove that there exists a positive integer $n$ and integers $a_{1}, a_{2}, \ldots, a_{n}=a, b_{1}, b_{2}, \ldots, b_{n}=b, c_{1}, c_{2}, \ldots, c_{n}=c$ such that for all $1 \leq i \leq n, a_{i-1} a_{i}+b_{i-1} b_{i}+c_{i-1} c_{i}=1$.