Let $v =(r,s,t) \in \mathbb{N}^3$ be a vector such that $\gcd(r,s,t)=1$. We know that there are vectors $x= (x_1,x_2,x_3) \in \mathbb{Z}^3 $ such that $v.x =1$. For each $v$, let $O(v)$ be the smallest size ($L^1$ or $L^\infty$ norm) of such $x$. Now for a natural number $n$ define $f(n)$ to be the maximum of $O(v)$, where $v$ ranges over vectors with $L^1$ norm at most n (i.e. , $r+s+t \leq n$). What is the asymptotic of $f(n)$?

Also similar question when the vector has a fixed number of entries (at least 3).